Preface to the First Edition xv

Preface to the Second Edition xix

Preface to the Third Edition xxi

1 Introduction 1

1.1 Fundamental Issues 2

1.2 Describing the System 3

1.3 Fundamental Forces 3

1.4 The Dynamical Equation 5

1.5 Solving the Dynamical Equation 7

1.6 Separation of Variables 8

1.6.1 Separating Space and Time Variables 9

1.6.2 Separating Nuclear and Electronic Variables 9

1.6.3 Separating Variables in General 10

1.7 Classical Mechanics 11

1.7.1 The Sun-Earth System 11

1.7.2 The Solar System 12

1.8 Quantum Mechanics 13

1.8.1 A Hydrogen-Like Atom 13

1.8.2 The Helium Atom 16

1.9 Chemistry 18

References 19

2 Force Field Methods 20

2.1 Introduction 20

2.2 The Force Field Energy 21

2.2.1 The Stretch Energy 23

2.2.2 The Bending Energy 25

2.2.3 The Out-of-Plane Bending Energy 28

2.2.4 The Torsional Energy 28

2.2.5 The van der Waals energy 32

2.2.6 The Electrostatic Energy: Atomic Charges 37

2.2.7 The Electrostatic Energy: Atomic Multipoles 41

2.2.8 The Electrostatic Energy: Polarizability and Charge Penetration Effects 42

2.2.9 Cross Terms 48

2.2.10 Small Rings and Conjugated Systems 49

2.2.11 Comparing Energies of Structurally Different Molecules 51

2.3 Force Field Parameterization 53

2.3.1 Parameter Reductions in Force Fields 58

2.3.2 Force Fields for Metal Coordination Compounds 59

2.3.3 Universal Force Fields 62

2.4 Differences in Atomistic Force Fields 62

2.5 Water Models 66

2.6 Coarse Grained Force Fields 67

2.7 Computational Considerations 69

2.8 Validation of Force Fields 71

2.9 Practical Considerations 73

2.10 Advantages and Limitations of Force Field Methods 73

2.11 Transition Structure Modeling 74

2.11.1 Modeling the TS as a Minimum Energy Structure 74

2.11.2 Modeling the TS as a Minimum Energy Structure on the Reactant/Product Energy Seam 75

2.11.3 Modeling the Reactive Energy Surface by Interacting Force Field Functions 76

2.11.4 Reactive Force Fields 77

2.12 Hybrid Force Field Electronic Structure Methods 78

References 82

3 Hartree-Fock Theory 88

3.1 The Adiabatic and Born-Oppenheimer Approximations 90

3.2 Hartree-Fock Theory 94

3.3 The Energy of a Slater Determinant 95

3.4 Koopmans' Theorem 100

3.5 The Basis Set Approximation 101

3.6 An Alternative Formulation of the Variational Problem 105

3.7 Restricted and Unrestricted Hartree-Fock 106

3.8 SCF Techniques 108

3.8.1 SCF Convergence 108

3.8.2 Use of Symmetry 110

3.8.3 Ensuring that the HF Energy Is a Minimum, and the Correct Minimum 111

3.8.4 Initial Guess Orbitals 113

3.8.5 Direct SCF 113

3.8.6 Reduced Scaling Techniques 116

3.8.7 Reduced Prefactor Methods 117

3.9 Periodic Systems 119

References 121

4 Electron Correlation Methods 124

4.1 Excited Slater Determinants 125

4.2 Configuration Interaction 128

4.2.1 ci Matrix Elements 129

4.2.2 Size of the CI Matrix 131

4.2.3 Truncated CI Methods 133

4.2.4 Direct CI Methods 134

4.3 Illustrating how CI Accounts for Electron Correlation, and the RHF Dissociation Problem 135

4.4 The UHF Dissociation and the Spin Contamination Problem 138

4.5 Size Consistency and Size Extensivity 142

4.6 Multiconfiguration Self-Consistent Field 143

4.7 Multireference Configuration Interaction 148

4.8 Many-Body Perturbation Theory 148

4.8.1 Møller-Plesset Perturbation Theory 151

4.8.2 Unrestricted and Projected Møller-Plesset Methods 156

4.9 Coupled Cluster 157

4.9.1 Truncated coupled cluster methods 160

4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 162

4.10.1 Illustrating Correlation Methods for the Beryllium Atom 165

4.11 Methods Involving the Interelectronic Distance 166

4.12 Techniques for Improving the Computational Efficiency 169

4.12.1 Direct Methods 170

4.12.2 Localized Orbital Methods 172

4.12.3 Fragment-Based Methods 173

4.12.4 Tensor Decomposition Methods 173

4.13 Summary of Electron Correlation Methods 174

4.14 Excited States 176

4.14.1 Excited State Analysis 181

4.15 Quantum Monte Carlo Methods 183

References 185

5 Basis Sets 188

5.1 Slater- and Gaussian-Type Orbitals 189

5.2 Classification of Basis Sets 190

5.3 Construction of Basis Sets 194

5.3.1 Exponents of Primitive Functions 194

5.3.2 Parameterized Exponent Basis Sets 195

5.3.3 Basis Set Contraction 196

5.3.4 Basis Set Augmentation 199

5.4 Examples of Standard Basis Sets 200

5.4.1 Pople Style Basis Sets 200

5.4.2 Dunning-Huzinaga Basis Sets 202

5.4.3 Karlsruhe-Type Basis Sets 203

5.4.4 Atomic Natural Orbital Basis Sets 203

5.4.5 Correlation Consistent Basis Sets 204

5.4.6 Polarization Consistent Basis Sets 205

5.4.7 Correlation Consistent F12 Basis Sets 206

5.4.8 Relativistic Basis Sets 207

5.4.9 Property Optimized Basis Sets 207

5.5 Plane Wave Basis Functions 208

5.6 Grid and Wavelet Basis Sets 210

5.7 Fitting Basis Sets 211

5.8 Computational Issues 211

5.9 Basis Set Extrapolation 212

5.10 Composite Extrapolation Procedures 215

5.10.1 Gaussian-n Models 216

5.10.2 Complete Basis Set Models 217

5.10.3 Weizmann-n Models 219

5.10.4 Other Composite Models 221

5.11 Isogyric and Isodesmic Reactions 222

5.12 Effective Core Potentials 223

5.13 Basis Set Superposition and Incompleteness Errors 226

References 228

6 Density Functional Methods 233

6.1 Orbital-Free Density Functional Theory 234

6.2 Kohn-Sham Theory 235

6.3 Reduced Density Matrix and Density Cumulant Methods 237

6.4 Exchange and Correlation Holes 241

6.5 Exchange-Correlation Functionals 244

6.5.1 Local Density Approximation 247

6.5.2 Generalized Gradient Approximation 248

6.5.3 Meta-GGA Methods 251

6.5.4 Hybrid or Hyper-GGA Methods 252

6.5.5 Double Hybrid Methods 253

6.5.6 Range-Separated Methods 254

6.5.7 Dispersion-Corrected Methods 255

6.5.8 Functional Overview 257

6.6 Performance of Density Functional Methods 258

6.7 Computational Considerations 260

6.8 Differences between Density Functional Theory and Hartree-Fock 262

6.9 Time-Dependent Density Functional Theory (TDDFT) 263

6.9.1 Weak Perturbation - Linear Response 266

6.10 Ensemble Density Functional Theory 268

6.11 Density Functional Theory Problems 269

6.12 Final Considerations 269

References 270

7 Semi-empirical Methods 275

7.1 Neglect of Diatomic Differential Overlap (NDDO) Approximation 276

7.2 Intermediate Neglect of Differential Overlap (INDO) Approximation 277

7.3 Complete Neglect of Differential Overlap (CNDO) Approximation 277

7.4 Parameterization 278

7.4.1 Modified Intermediate Neglect of Differential Overlap (MINDO) 278

7.4.2 Modified NDDO Models 279

7.4.3 Modified Neglect of Diatomic Overlap (MNDO) 280

7.4.4 Austin Model 1 (AM1) 281

7.4.5 Modified Neglect of Diatomic Overlap, Parametric Method Number 3 (PM3) 281

7.4.6 The MNDO/d and AM1/d Methods 282

7.4.7 Parametric Method Numbers 6 and 7 (PM6 and PM7) 282

7.4.8 Orthogonalization Models 283

7.5 Hückel Theory 283

7.5.1 Extended Hückel theory 283

7.5.2 Simple Hückel Theory 284

7.6 Tight-Binding Density Functional Theory 285

7.7 Performance of Semi-empirical Methods 287

7.8 Advantages and Limitations of Semi-empirical Methods 289

References 290

8 Valence Bond Methods 291

8.1 Classical Valence Bond Theory 292

8.2 Spin-Coupled Valence Bond Theory 293

8.3 Generalized Valence Bond Theory 297

References 298

9 Relativistic Methods 299

9.1 The Dirac Equation 300

9.2 Connections between the Dirac and Schrödinger Equations 302

9.2.1 Including Electric Potentials 302

9.2.2 Including Both Electric and Magnetic Potentials 304

9.3 Many-Particle Systems 306

9.4 Four-Component Calculations 309

9.5 Two-Component Calculations 310

9.6 Relativistic Effects 313

References 315

10 Wave Function Analysis 317

10.1 Population Analysis Based on Basis Functions 317

10.2 Population Analysis Based on the Electrostatic Potential 320

10.3 Population Analysis Based on the Electron Density 323

10.3.1 Quantum Theory of Atoms in Molecules 324

10.3.2 Voronoi, Hirshfeld, Stockholder and Stewart Atomic Charges 327

10.3.3 Generalized Atomic Polar Tensor Charges 329

10.4 Localized Orbitals 329

10.4.1 Computational considerations 332

10.5 Natural Orbitals 333

10.5.1 Natural Atomic Orbital and Natural Bond Orbital Analyses 334

10.6 Computational Considerations 337

10.7 Examples 338

References 339

11 Molecular Properties 341

11.1 Examples of Molecular Properties 343

11.1.1 External Electric Field 343

11.1.2 External Magnetic Field 344

11.1.3 Nuclear Magnetic Moments 345

11.1.4 Electron Magnetic Moments 345

11.1.5 Geometry Change 346

11.1.6 Mixed Derivatives 346

11.2 Perturbation Methods 347

11.3 Derivative Techniques 349