Introduction to Computational Chemistry

Professor Frank Jensen, Department of Chemistry, Aarhus University, DenmarkFrank Jensen obtained his Ph.D. from UCLA in 1987 with Professors C. S. Foote and K. N. Houk, and is currently an Associate Professor in the Department of Chemistry, Aarhus University, Denmark. He has published over 120 papers and articles, and has been a member of the editorial boards of Advances in Quantum Chemistry (2005 - 2011) and the International Journal of Quantum Chemistry (2006-2011).

• Preface to the First Edition xvPreface to the Second Edition xixPreface to the Third Edition xxi1 Introduction 11.1 Fundamental Issues 21.2 Describing the System 31.3 Fundamental Forces 31.4 The Dynamical Equation 51.5 Solving the Dynamical Equation 71.6 Separation of Variables 81.6.1 Separating Space and Time Variables 91.6.2 Separating Nuclear and Electronic Variables 91.6.3 Separating Variables in General 101.7 Classical Mechanics 111.7.1 The Sun–Earth System 111.7.2 The Solar System 121.8 Quantum Mechanics 131.8.1 A Hydrogen-Like Atom 131.8.2 The Helium Atom 161.9 Chemistry 18References 192 Force Field Methods 202.1 Introduction 202.2 The Force Field Energy 212.2.1 The Stretch Energy 232.2.2 The Bending Energy 252.2.3 The Out-of-Plane Bending Energy 282.2.4 The Torsional Energy 282.2.5 The van der Waals energy 322.2.6 The Electrostatic Energy: Atomic Charges 372.2.7 The Electrostatic Energy: Atomic Multipoles 412.2.8 The Electrostatic Energy: Polarizability and Charge Penetration Effects 422.2.9 Cross Terms 482.2.10 Small Rings and Conjugated Systems 492.2.11 Comparing Energies of Structurally Different Molecules 512.3 Force Field Parameterization 532.3.1 Parameter Reductions in Force Fields 582.3.2 Force Fields for Metal Coordination Compounds 592.3.3 Universal Force Fields 622.4 Differences in Atomistic Force Fields 622.5 Water Models 662.6 Coarse Grained Force Fields 672.7 Computational Considerations 692.8 Validation of Force Fields 712.9 Practical Considerations 732.10 Advantages and Limitations of Force Field Methods 732.11 Transition Structure Modeling 742.11.1 Modeling the TS as a Minimum Energy Structure 742.11.2 Modeling the TS as a Minimum Energy Structure on the Reactant/Product Energy Seam 752.11.3 Modeling the Reactive Energy Surface by Interacting Force Field Functions 762.11.4 Reactive Force Fields 772.12 Hybrid Force Field Electronic Structure Methods 78References 823 Hartree–Fock Theory 883.1 The Adiabatic and Born–Oppenheimer Approximations 903.2 Hartree–Fock Theory 943.3 The Energy of a Slater Determinant 953.4 Koopmans’ Theorem 1003.5 The Basis Set Approximation 1013.6 An Alternative Formulation of the Variational Problem 1053.7 Restricted and Unrestricted Hartree–Fock 1063.8 SCF Techniques 1083.8.1 SCF Convergence 1083.8.2 Use of Symmetry 1103.8.3 Ensuring that the HF Energy Is a Minimum, and the Correct Minimum 1113.8.4 Initial Guess Orbitals 1133.8.5 Direct SCF 1133.8.6 Reduced Scaling Techniques 1163.8.7 Reduced Prefactor Methods 1173.9 Periodic Systems 119References 1214 Electron Correlation Methods 1244.1 Excited Slater Determinants 1254.2 Configuration Interaction 1284.2.1 ci Matrix Elements 1294.2.2 Size of the CI Matrix 1314.2.3 Truncated CI Methods 1334.2.4 Direct CI Methods 1344.3 Illustrating how CI Accounts for Electron Correlation, and the RHF Dissociation Problem 1354.4 The UHF Dissociation and the Spin Contamination Problem 1384.5 Size Consistency and Size Extensivity 1424.6 Multiconfiguration Self-Consistent Field 1434.7 Multireference Configuration Interaction 1484.8 Many-Body Perturbation Theory 1484.8.1 Møller–Plesset Perturbation Theory 1514.8.2 Unrestricted and Projected Møller–Plesset Methods 1564.9 Coupled Cluster 1574.9.1 Truncated coupled cluster methods 1604.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 1624.10.1 Illustrating Correlation Methods for the Beryllium Atom 1654.11 Methods Involving the Interelectronic Distance 1664.12 Techniques for Improving the Computational Efficiency 1694.12.1 Direct Methods 1704.12.2 Localized Orbital Methods 1724.12.3 Fragment-Based Methods 1734.12.4 Tensor Decomposition Methods 1734.13 Summary of Electron Correlation Methods 1744.14 Excited States 1764.14.1 Excited State Analysis 1814.15 Quantum Monte Carlo Methods 183References 1855 Basis Sets 1885.1 Slater- and Gaussian-Type Orbitals 1895.2 Classification of Basis Sets 1905.3 Construction of Basis Sets 1945.3.1 Exponents of Primitive Functions 1945.3.2 Parameterized Exponent Basis Sets 1955.3.3 Basis Set Contraction 1965.3.4 Basis Set Augmentation 1995.4 Examples of Standard Basis Sets 2005.4.1 Pople Style Basis Sets 2005.4.2 Dunning–Huzinaga Basis Sets 2025.4.3 Karlsruhe-Type Basis Sets 2035.4.4 Atomic Natural Orbital Basis Sets 2035.4.5 Correlation Consistent Basis Sets 2045.4.6 Polarization Consistent Basis Sets 2055.4.7 Correlation Consistent F12 Basis Sets 2065.4.8 Relativistic Basis Sets 2075.4.9 Property Optimized Basis Sets 2075.5 Plane Wave Basis Functions 2085.6 Grid and Wavelet Basis Sets 2105.7 Fitting Basis Sets 2115.8 Computational Issues 2115.9 Basis Set Extrapolation 2125.10 Composite Extrapolation Procedures 2155.10.1 Gaussian-n Models 2165.10.2 Complete Basis Set Models 2175.10.3 Weizmann-n Models 2195.10.4 Other Composite Models 2215.11 Isogyric and Isodesmic Reactions 2225.12 Effective Core Potentials 2235.13 Basis Set Superposition and Incompleteness Errors 226References 2286 Density Functional Methods 2336.1 Orbital-Free Density Functional Theory 2346.2 Kohn–Sham Theory 2356.3 Reduced Density Matrix and Density Cumulant Methods 2376.4 Exchange and Correlation Holes 2416.5 Exchange–Correlation Functionals 2446.5.1 Local Density Approximation 2476.5.2 Generalized Gradient Approximation 2486.5.3 Meta-GGA Methods 2516.5.4 Hybrid or Hyper-GGA Methods 2526.5.5 Double Hybrid Methods 2536.5.6 Range-Separated Methods 2546.5.7 Dispersion-Corrected Methods 2556.5.8 Functional Overview 2576.6 Performance of Density Functional Methods 2586.7 Computational Considerations 2606.8 Differences between Density Functional Theory and Hartree-Fock 2626.9 Time-Dependent Density Functional Theory (TDDFT) 2636.9.1 Weak Perturbation – Linear Response 2666.10 Ensemble Density Functional Theory 2686.11 Density Functional Theory Problems 2696.12 Final Considerations 269References 2707 Semi-empirical Methods 2757.1 Neglect of Diatomic Differential Overlap (NDDO) Approximation 2767.2 Intermediate Neglect of Differential Overlap (INDO) Approximation 2777.3 Complete Neglect of Differential Overlap (CNDO) Approximation 2777.4 Parameterization 2787.4.1 Modified Intermediate Neglect of Differential Overlap (MINDO) 2787.4.2 Modified NDDO Models 2797.4.3 Modified Neglect of Diatomic Overlap (MNDO) 2807.4.4 Austin Model 1 (AM1) 2817.4.5 Modified Neglect of Diatomic Overlap, Parametric Method Number 3 (PM3) 2817.4.6 The MNDO/d and AM1/d Methods 2827.4.7 Parametric Method Numbers 6 and 7 (PM6 and PM7) 2827.4.8 Orthogonalization Models 2837.5 Hückel Theory 2837.5.1 Extended Hückel theory 2837.5.2 Simple Hückel Theory 2847.6 Tight-Binding Density Functional Theory 2857.7 Performance of Semi-empirical Methods 2877.8 Advantages and Limitations of Semi-empirical Methods 289References 2908 Valence Bond Methods 2918.1 Classical Valence Bond Theory 2928.2 Spin-Coupled Valence Bond Theory 2938.3 Generalized Valence Bond Theory 297References 2989 Relativistic Methods 2999.1 The Dirac Equation 3009.2 Connections between the Dirac and Schrödinger Equations 3029.2.1 Including Electric Potentials 3029.2.2 Including Both Electric and Magnetic Potentials 3049.3 Many-Particle Systems 3069.4 Four-Component Calculations 3099.5 Two-Component Calculations 3109.6 Relativistic Effects 313References 31510 Wave Function Analysis 31710.1 Population Analysis Based on Basis Functions 31710.2 Population Analysis Based on the Electrostatic Potential 32010.3 Population Analysis Based on the Electron Density 32310.3.1 Quantum Theory of Atoms in Molecules 32410.3.2 Voronoi, Hirshfeld, Stockholder and Stewart Atomic Charges 32710.3.3 Generalized Atomic Polar Tensor Charges 32910.4 Localized Orbitals 32910.4.1 Computational considerations 33210.5 Natural Orbitals 33310.5.1 Natural Atomic Orbital and Natural Bond Orbital Analyses 33410.6 Computational Considerations 33710.7 Examples 338References 33911 Molecular Properties 34111.1 Examples of Molecular Properties 34311.1.1 External Electric Field 34311.1.2 External Magnetic Field 34411.1.3 Nuclear Magnetic Moments 34511.1.4 Electron Magnetic Moments 34511.1.5 Geometry Change 34611.1.6 Mixed Derivatives 34611.2 Perturbation Methods 34711.3 Derivative Techniques 34911.4 Response and Propagator Methods 35111.5 Lagrangian Techniques 35111.6 Wave Function Response 35311.6.1 Coupled Perturbed Hartree–Fock 35411.7 Electric Field Perturbation 35711.7.1 External Electric Field 35711.7.2 Internal Electric Field 35811.8 Magnetic Field Perturbation 35811.8.1 External Magnetic Field 36011.8.2 Nuclear Spin 36111.8.3 Electron Spin 36111.8.4 Electron Angular Momentum 36211.8.5 Classical Terms 36211.8.6 Relativistic Terms 36311.8.7 Magnetic Properties 36311.8.8 Gauge Dependence of Magnetic Properties 36611.9 Geometry Perturbations 36711.10 Time-Dependent Perturbations 37211.11 Rotational and Vibrational Corrections 37711.12 Environmental Effects 37811.13 Relativistic Corrections 378References 37812 Illustrating the Concepts 38012.1 Geometry Convergence 38012.1.1 Wave Function Methods 38012.1.2 Density Functional Methods 38212.2 Total Energy Convergence 38312.3 Dipole Moment Convergence 38512.3.1 Wave Function Methods 38512.3.2 Density Functional Methods 38512.4 Vibrational Frequency Convergence 38612.4.1 Wave Function Methods 38612.5 Bond Dissociation Curves 38912.5.1 Wave Function Methods 38912.5.2 Density Functional Methods 39412.6 Angle Bending Curves 39412.7 Problematic Systems 39612.7.1 The Geometry of FOOF 39612.7.2 The Dipole Moment of CO 39712.7.3 The Vibrational Frequencies of O3 39812.8 Relative Energies of C4H6 Isomers 399References 40213 Optimization Techniques 40413.1 Optimizing Quadratic Functions 40513.2 Optimizing General Functions: Finding Minima 40713.2.1 Steepest Descent 40713.2.2 Conjugate Gradient Methods 40813.2.3 Newton–Raphson Methods 40913.2.4 Augmented Hessian Methods 41013.2.5 Hessian Update Methods 41113.2.6 Truncated Hessian Methods 41313.2.7 Extrapolation: The DIIS Method 41313.3 Choice of Coordinates 41513.4 Optimizing General Functions: Finding Saddle Points (Transition Structures) 41813.4.1 One-Structure Interpolation Methods 41913.4.2 Two-Structure Interpolation Methods 42113.4.3 Multistructure Interpolation Methods 42213.4.4 Characteristics of Interpolation Methods 42613.4.5 Local Methods: Gradient Norm Minimization 42713.4.6 Local Methods: Newton–Raphson 42713.4.7 Local Methods: The Dimer Method 42913.4.8 Coordinates for TS Searches 42913.4.9 Characteristics of Local Methods 43013.4.10 Dynamic Methods 43113.5 Constrained Optimizations 43113.6 Global Minimizations and Sampling 43313.6.1 Stochastic and Monte Carlo Methods 43413.6.2 Molecular Dynamics Methods 43613.6.3 Simulated Annealing 43613.6.4 Genetic Algorithms 43713.6.5 Particle Swarm and Gravitational Search Methods 43713.6.6 Diffusion Methods 43813.6.7 Distance Geometry Methods 43913.6.8 Characteristics of Global Optimization Methods 43913.7 Molecular Docking 44013.8 Intrinsic Reaction Coordinate Methods 441References 44414 Statistical Mechanics and Transition State Theory 44714.1 Transition State Theory 44714.2 Rice–Ramsperger–Kassel–Marcus Theory 45014.3 Dynamical Effects 45114.4 Statistical Mechanics 45214.5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation 45414.5.1 Translational Degrees of Freedom 45514.5.2 Rotational Degrees of Freedom 45514.5.3 Vibrational Degrees of Freedom 45714.5.4 Electronic Degrees of Freedom 45814.5.5 Enthalpy and Entropy Contributions 45914.6 Condensed Phases 464References 46815 Simulation Techniques 46915.1 Monte Carlo Methods 47215.1.1 Generating Non-natural Ensembles 47415.2 Time-Dependent Methods 47415.2.1 Molecular Dynamics Methods 47415.2.2 Generating Non-natural Ensembles 47815.2.3 Langevin Methods 47915.2.4 Direct Methods 47915.2.5 Ab Initio Molecular Dynamics 48015.2.6 Quantum Dynamical Methods Using Potential Energy Surfaces 48315.2.7 Reaction Path Methods 48415.2.8 Non-Born–Oppenheimer Methods 48715.2.9 Constrained and Biased Sampling Methods 48815.3 Periodic Boundary Conditions 49115.4 Extracting Information from Simulations 49415.5 Free Energy Methods 49915.5.1 Thermodynamic Perturbation Methods 49915.5.2 Thermodynamic Integration Methods 50015.6 Solvation Models 50215.6.1 Continuum Solvation Models 50315.6.2 Poisson–Boltzmann Methods 50515.6.3 Born/Onsager/Kirkwood Models 50615.6.4 Self-Consistent Reaction Field Models 508References 51116 Qualitative Theories 51516.1 Frontier Molecular Orbital Theory 51516.2 Concepts from Density Functional Theory 51916.3 Qualitative Molecular Orbital Theory 52216.4 Energy Decomposition Analyses 52416.5 Orbital Correlation Diagrams: The Woodward–Hoffmann Rules 52616.6 The Bell–Evans–Polanyi Principle/Hammond Postulate/Marcus Theory 53416.7 More O’Ferrall–Jencks Diagrams 538References 54117 Mathematical Methods 54317.1 Numbers, Vectors, Matrices and Tensors 54317.2 Change of Coordinate System 54917.2.1 Examples of Changing the Coordinate System 55417.2.2 Vibrational Normal Coordinates 55517.2.3 Energy of a Slater Determinant 55717.2.4 Energy of a CI Wave Function 55817.2.5 Computational Considerations 55817.3 Coordinates, Functions, Functionals, Operators and Superoperators 56017.3.1 Differential Operators 56217.4 Normalization, Orthogonalization and Projection 56317.5 Differential Equations 56517.5.1 Simple First-Order Differential Equations 56517.5.2 Less Simple First-Order Differential Equations 56617.5.3 Simple Second-Order Differential Equations 56617.5.4 Less Simple Second-Order Differential Equations 56717.5.5 Second-Order Differential Equations Depending on the Function Itself 56817.6 Approximating Functions 56817.6.1 Taylor Expansion 56917.6.2 Basis Set Expansion 57017.6.3 Tensor Decomposition Methods 57217.6.4 Examples of Tensor Decompositions 57417.7 Fourier and Laplace Transformations 57717.8 Surfaces 577References 58018 Statistics and QSAR 58118.1 Introduction 58118.2 Elementary Statistical Measures 58318.3 Correlation between Two Sets of Data 58518.4 Correlation between Many Sets of Data 58818.4.1 Quality Measures 58918.4.2 Multiple Linear Regression 59018.4.3 Principal Component Analysis 59118.4.4 Partial Least Squares 59318.4.5 Illustrative Example 59418.5 Quantitative Structure–Activity Relationships (QSAR) 59518.6 Non-linear Correlation Methods 59718.7 Clustering Methods 598References 60419 Concluding Remarks 605Appendix A 608Notation 608Appendix B 614The Variational Principle 614The Hohenberg–Kohn Theorems 615The Adiabatic Connection Formula 616Reference 617Appendix C 618Atomic Units 618Appendix D 619Z Matrix Construction 619Appendix E 627First and Second Quantization 627References 628Index 629