Contents

Preface to the First Edition xv

Preface to the Second Edition xix

1 Introduction 1

1.1 Fundamental Issues 2

1.2 Describing the System 3

1.3 Fundamental Forces 4

1.4 The Dynamical Equation 5

1.5 Solving the Dynamical Equation 8

1.6 Separation of Variables 8

1.6.1 Separating space and time variables 10

1.6.2 Separating nuclear and electronic variables 10

1.6.3 Separating variables in general 11

1.7 Classical Mechanics 12

1.7.1 The Sun-Earth system 12

1.7.2 The solar system 13

1.8 Quantum Mechanics 14

1.8.1 A hydrogen-like atom 14

1.8.2 The helium atom 17

1.9 Chemistry 19 References 21

2 Force Field Methods 22

2.1 Introduction 22

2.2 The Force Field Energy 24

2.2.1 The stretch energy 25

2.2.2 The bending energy 27

2.2.3 The out-of-plane bending energy 30

2.2.4 The torsional energy 30

2.2.5 The van der Waals energy 34

2.2.6 The electrostatic energy: charges and dipoles 40

2.2.7 The electrostatic energy: multipoles and polarizabilities 43

2.2.8 Cross terms 47

2.2.9 Small rings and conjugated systems 48

2.2.10 Comparing energies of structurally different molecules 50

2.3 Force Field Parameterization 51

2.3.1 Parameter reductions in force fields 57

2.3.2 Force fields for metal coordination compounds 58

2.3.3 Universal force fields 62

2.4 Differences in Force Fields 62

2.5 Computational Considerations 65

2.6 Validation of Force Fields 67

2.7 Practical Considerations 69

2.8 Advantages and Limitations of Force Field Methods 69

2.9 Transition Structure Modelling 70

2.9.1 Modelling the TS as a minimum energy structure 70

2.9.2 Modelling the TS as a minimum energy structure on the reactant/

product energy seam 71

2.9.3 Modelling the reactive energy surface by interacting force field functions or by geometry-dependent parameters 73

2.10 Hybrid Force Field Electronic Structure Methods 74 References 77

3 Electronic Structure Methods: Independent-Particle Models 80

3.1 The Adiabatic and Born-Oppenheimer Approximations 82

3.2 Self-Consistent Field Theory 86

3.3 The Energy of a Slater Determinant 87

3.4 Koopmans' Theorem 92

3.5 The Basis Set Approximation 93

3.6 An Alternative Formulation of the Variational Problem 98

3.7 Restricted and Unrestricted Hartree-Fock 99

3.8 SCF Techniques 100

3.8.1 SCF convergence 101

3.8.2 Use of symmetry 104

3.8.3 Ensuring that the HF energy is a minimum, and the correct minimum 105

3.8.4 Initial guess orbitals 107

3.8.5 Direct SCF 108

3.8.6 Reduced scaling techniques 110

3.9 Periodic Systems 113

3.10 Semi-Empirical Methods 115

3.10.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 116

3.10.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 117

3.10.3 Complete Neglect of Differential Overlap Approximation (CNDO) 117

3.11 Parameterization 118

3.11.1 Modified Intermediate Neglect of Differential Overlap (MINDO) 119

3.11.2 Modified NDDO models 119

3.11.3 Modified Neglect of Diatomic Overlap (MNDO) 121

3.11.4 Austin Model 1 (AM1) 121

3.11.5 Modified Neglect of Diatomic Overlap, Parametric Method Number 3 (PM3) 122

3.11.6 Parametric Method number 5 (PM5) and PDDG/PM3 methods 123

3.11.7 The MNDO/d and AM1/d methods 124

3.11.8 Semi Ab initio Method 1 124

3.12 Performance of Semi-Empirical Methods 125

3.13 Huckel Theory 127

3.13.1 Extended Huckel theory 127

3.13.2 Simple Huckel theory 128

3.14 Limitations and Advantages of Semi-Empirical Methods 129 References 131

4 Electron Correlation Methods 133

4.1 Excited Slater Determinants 135

4.2 Configuration Interaction 137

4.2.1 CI Matrix elements 138

4.2.2 Size of the CI matrix 141

4.2.3 Truncated CI methods 143

4.2.4 Direct CI methods 144

4.3 Illustrating how CI Accounts for Electron Correlation, and the

RHF Dissociation Problem 145

4.4 The UHF Dissociation, and the Spin Contamination Problem 148

4.5 Size Consistency and Size Extensivity 153

4.6 Multi-Configuration Self-Consistent Field 153

4.7 Multi-Reference Configuration Interaction 158

4.8 Many-Body Perturbation Theory 159

4.8.1 M0ller-Plesset perturbation theory 162

4.8.2 Unrestricted and projected M0ller-Plesset methods 168

4.9 Coupled Cluster 169 4.9.1 Truncated coupled cluster methods 172

4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 174

4.10.1 Illustrating correlation methods for the beryllium atom 177

4.11 Methods Involving the Interelectronic Distance 178

4.12 Direct Methods 181

4.13 Localized Orbital Methods 182

4.14 Summary of Electron Correlation Methods 183

4.15 Excited States 186

4.16 Quantum Monte Carlo Methods 187 References 189

5 Basis Sets 192

5.1 Slater and Gaussian Type Orbitals 192

5.2 Classification of Basis Sets 194

5.3 Even- and Well-Tempered Basis Sets 198

5.4 Contracted Basis Sets 200

5.4.1 Pople style basis sets 202

5.4.2 Dunning-Huzinaga basis sets 204

5.4.3 MINI, MIDI and MAXI basis sets 205

5.4.4 Ahlrichs type basis sets 205

5.4.5 Atomic natural orbital basis sets 205

5.4.6 Correlation consistent basis sets 206

5.4.7 Polarization consistent basis sets 207

5.4.8 Basis set extrapolation 208

5.5 Plane Wave Basis Functions 211

5.6 Recent Developments and Computational Issues 212

5.7 Composite Extrapolation Procedures 213

5.8 Isogyric and Isodesmic Reactions 221

5.9 Effective Core Potentials 222

5.10 Basis Set Superposition Errors 225

5.11 Pseudospectral Methods 227 References 229

6 Density Functional Methods 232

6.1 Orbital-Free Density Functional Theory 233

6.2 Kohn-Sham Theory 235

6.3 Reduced Density Matrix Methods 236

6.4 Exchange and Correlation Holes 240

6.5 Exchange-Correlation Functionals 243

6.5.1 Local Density Approximation 246

6.5.2 Gradient-corrected methods 248

6.5.3 Higher order gradient or meta-GGA methods 250

6.5.4 Hybrid or hyper-GGA methods 252

6.5.5 Generalized random phase methods 253

6.5.6 Functionals overview 254

6.6 Performance and Properties of Density Functional Methods 255

6.7 DFT Problems 258

6.8 Computational Considerations 260

6.9 Final Considerations 263 References 264

7 Valence Bond Methods 268

7.1 Classical Valence Bond Theory 269

7.2 Spin-Coupled Valence Bond Theory 270

7.3 Generalized Valence Bond Theory 275 References 276

8 Relativistic Methods 277

8.1 The Dirac Equation 278

8.2 Connections Between the Dirac and Schrodinger Equations 280

8.2.1 Including electric potentials 280

8.2.2 Including both electric and magnetic potentials 282

8.3 Many-Particle Systems 284

8.4 Four-Component Calculations 287

8.5 Relativistic Effects 289 References 292

9 Wave Function Analysis 293

9.1 Population Analysis Based on Basis Functions 293

9.2 Population Analysis Based on the Electrostatic Potential 296

9.3 Population Analysis Based on the Electron Density 299

9.3.1 Atoms In Molecules 299

9.3.2 Voronoi, Hirshfeld and Stewart atomic charges 303

9.3.3 Generalized atomic polar tensor charges 304

9.4 Localized Orbitals 304 9.4.1 Computational considerations 306

9.5 Natural Orbitals 308

9.6 Natural Atomic Orbital and Natural Bond Orbital Analysis 309

9.7 Computational Considerations 311

9.8 Examples 312 References 313

10 Molecular Properties 315

10.1 Examples of Molecular Properties 316

10.1.1 External electric field 316

10.1.2 External magnetic field 318

10.1.3 Internal magnetic moments 318

10.1.4 Geometry change 319

10.1.5 Mixed derivatives 319

10.2 Perturbation Methods 321

10.3 Derivative Techniques 321

10.4 Lagrangian Techniques 324

10.5 Coupled Perturbed Hartree-Fock 325

10.6 Electric Field Perturbation 329

10.6.1 External electric field 329

10.6.2 Internal electric field 329

10.7 Magnetic Field Perturbation 329

10.7.1 External magnetic field 331

10.7.2 Nuclear spin 332

10.7.3 Electron spin 333

10.7.4 Classical terms 333

10.7.5 Relativistic terms 334

10.7.6 Magnetic properties 334

10.7.7 Gauge dependence of magnetic properties 338

10.8 Geometry Perturbations 339

10.9 Response and Propagator Methods 343

10.10 Property Basis Sets 348 References 349

11 Illustrating the Concepts 350

11.1 Geometry Convergence 350

11.1.1 Ab Initio methods 350

11.1.2 Density functional methods 353

11.2 Total Energy Convergence 354

11.3 Dipole Moment Convergence 356

11.3.1 Ab Initio methods 356

11.3.2 Density functional methods 357

11.4 Vibrational Frequency Convergence 358

11.4.1 Ab Initio methods 358

11.4.2 Density functional methods 360

11.5 Bond Dissociation Curves 361

11.5.1 Basis set effect at the Hartree-Fock level 361

11.5.2 Performance of different types of wave function 363

11.5.3 Density functional methods 369

11.6 Angle Bending Curves 370

11.7 Problematic Systems 370

11.7.1 The geometry of FOOF 371

11.7.2 The dipole moment of CO 372

11.7.3 The vibrational frequencies of O3 373

11.8 Relative Energies of C4H6 Isomers 374 References 378

12 Optimization Techniques 380

12.1 Optimizing Quadratic Functions 381

12.2 Optimizing General Functions: Finding Minima 383

12.2.1 Steepest descent 383

12.2.2 Conjugate gradient methods 384

12.2.3 Newton-Raphson methods 385

12.2.4 Step control 386

12.2.5 Obtaining the Hessian 387

12.2.6 Storing and diagonalizing the Hessian 388

12.2.7 Extrapolations: the GDIIS method 389

12.3 Choice of Coordinates 390

12.4 Optimizing General Functions: Finding Saddle Points (Transition Structures) 394

12.4.1 One-structure interpolation methods: coordinate driving, linear and quadratic synchronous transit, and sphere optimization

12.4.2 Two-structure interpolation methods: saddle, line-then-plane, ridge and step-and-slide optimizations

12.4.3 Multi-structure interpolation methods: chain, locally updated planes, self-penalty walk, conjugate peak refinement and nudged elastic band

12.4.4 Characteristics of interpolation methods

12.4.5 Local methods: gradient norm minimization

12.4.6 Local methods: Newton-Raphson

12.4.7 Local methods: the dimer method

12.4.8 Coordinates for TS searches

12.4.9 Characteristics of local methods

12.4.10 Dynamic methods

12.5 Constrained Optimization Problems

12.6 Conformational Sampling and the Global Minimum Problem

12.6.1 Stochastic and Monte Carlo methods

12.6.2 Molecular dynamics

12.6.3 Simulated annealing

12.6.4 Genetic algorithms

12.6.5 Diffusion methods

12.6.6 Distance geometry methods

12.7 Molecular Docking

12.8 Intrinsic Reaction Coordinate Methods References

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