The Basics of Theoretical and Computational Chemistry
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1 Bernd M. Rode, Thomas S. Hofer, and Michael D. Kugler The Basics of Theoretical and Computational Chemistry BICENTENNIA BICENTBNN I AL. WILEY-VCH Verlag GmbH & Co. KGaA
2 V Contents Preface IX 1 Introduction Theory and Models - Interpretation of Experimental Data The Notation Vector Space V" and Function Space F" 5 The Scalar Product Linear Transformation - Change of Basis Normalisation and Orthogonalisation of Vectors Matrix Representation of the Scalar Product Dual Vector Space and Hubert Space Probability Concept and the *P Function Operators Representation of Operators in a Basis Change of Basis in Representations of Operators 11 Test Questions Related to this Chapter 12 2 Basic Concepts of Vector Space Theory of Matter The Wave Equation as Probability Function The Postulates of Quantum Mechanics The Schrödinger Equation Hermicity Exact Measurability and Eigenvalue Problems Eigenvalue Problem of Hermitian Operators The Eigenvalue Equation of the Hamiltonian Eigenvalue Spectrum 23 Test Questions Related to this Chapter 24 3 Consequences of Quantum Mechanics Geometrical Interpretation of Eigenvalue Equations in Vector Space Commutators and Uncertainty Relationships Virtual Particles and the Forces in Nature 28 Test Questions Related to this Chapter 29 The Basics oftheoretical and Computational Oiemistry. Edited by B. M. Rode, T. S. Hofer and M. D. Kugler Copyright 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN:
3 Chemistry and Quantum Mechanics 31 1 Eigenvalue Problem of Angular Momentum and 'Orbital' Concept 31 2 Molecular Orbital and Valence Bond Models 32 3 Spin and the Antisymmetry Principle 35 4 The Virial Theorem 36 5 The Chemical Bond General Considerations and One-Electron Contributions Chemical Bonds in n-electron Systems Qualitative MO Models for Molecules 44 Test Questions Related to this Chapter 45 Approximations for Many-Electron Systems 47 1 Non-Relativistic Stationary Systems 47 2 Adiabatic Approximation - The Born-Oppenheimer Approximation 48 3 The Independent Particle Approximation 49 4 Spin Orbitals and Slater Determinants 50 5 Atomic and Molecular Orbitals: The LCAO-MO Approach 52 6 Quantitative Molecular Orbital Calculations Calculations with Slater Determinants Overlap Integrals Integrals of One-Electron Operators Integrals of Two-Electron Operators The Hartree-Fock Method Hartree-Fock Calculations in the LCAO-MO Approach: The Roothaan- Hall Equation 58 7 Canonical and Localised Molecular Orbitals and Chemical Model Concepts 64 Test Questions Related to this Chapter 71 Perturbation Theory in Quantum Chemistry 73 1 Projections and Projectors 73 2 Principles of Perturbation Theory 75 3 The Rayleigh-Schrödinger Perturbation Method 77 4 Applications of Perturbation Theory in Quantum Chemistry 79 Test Questions Related to this Chapter 80 Croup Theory in Theoretical Chemistry 81 1 Definition of a Group 81 2 Symmetry Groups Symmetry Operators Symmetry Groups and their Representations Reducible and Irreducible Representations and Character Tables 86 3 Applications of Group Theory in Quantum Chemistry 93 4 Applications of Group Theory in Spectroscopy Example 1: Electron Spectroscopy 95
4 Contents VII Example 2: Infrared/Raman Spectroscopy 96 Test Questions Related to this Chapter Computational Quantum Chemistry Methods Ab Initio Methods Ab Initio Hartree-Fock (HF) Methods Ab Initio Correlated Methods Configuration Interaction Methods Multi-Configüration Methods Coupled Cluster Methods Pair Methods Perturbational Methods Semiempirical MO Methods Density Functional Methods Local Density Approximation (LDA) Generalised Gradient Approximation (GGA) Hybrid Functionals 122 Test Questions Related to this Chapter Force Field Methods and Molecular Modelling Empirical Force Fields Molecular Modelling Programs Docking Quantitative Structure-Activity Relationships (QSARs) Multivariate Linear Regression (MLR) Nonlinear Regression Alternate Conditional Expectations (ACE) Project Pursuit Regression (PPR) Multivariate Adaptive Regression Splines (MARS) Example Calculation 134 Test Questions Related to this Chapter Statistical Simulations: Monte Carlo and Molecular Dynamics Methods Common Features Monte Carlo Simulations Molecular Dynamics Simulations Evaluation and Visualisation of Simulation Results Structure Dynamics Specific Evaluations in Macromolecule Simulations Quantum Mechanical Simulations Ab initio QM/MM Simulations Car-Parrinello DFT Simulations 165 Test Questions Related to this Chapter 166
5 VIII Contents 11 Synopsis 167 Appendix 1 Ab Initio Hartree-Fock Caiculations for Hyposulfuric Acid (H2SO3), including Optimisation of the Ceometry 169 Appendix 2 Books Recommended for Further Reading 173 Index 175
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