Introduction to Computational Chemistry

Professor Frank Jensen, Department of Chemistry, Aarhus University, Denmark Frank 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 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

11.4 Response and Propagator Methods 351

11.5 Lagrangian Techniques 351

11.6 Wave Function Response 353

11.6.1 Coupled Perturbed Hartree–Fock 354

11.7 Electric Field Perturbation 357

11.7.1 External Electric Field 357

11.7.2 Internal Electric Field 358

11.8 Magnetic Field Perturbation 358

11.8.1 External Magnetic Field 360

11.8.2 Nuclear Spin 361

11.8.3 Electron Spin 361

11.8.4 Electron Angular Momentum 362

11.8.5 Classical Terms 362

11.8.6 Relativistic Terms 363

11.8.7 Magnetic Properties 363

11.8.8 Gauge Dependence of Magnetic Properties 366

11.9 Geometry Perturbations 367

11.10 Time-Dependent Perturbations 372

11.11 Rotational and Vibrational Corrections 377

11.12 Environmental Effects 378

11.13 Relativistic Corrections 378

References 378

12 Illustrating the Concepts 380

12.1 Geometry Convergence 380

12.1.1 Wave Function Methods 380

12.1.2 Density Functional Methods 382

12.2 Total Energy Convergence 383

12.3 Dipole Moment Convergence 385

12.3.1 Wave Function Methods 385

12.3.2 Density Functional Methods 385

12.4 Vibrational Frequency Convergence 386

12.4.1 Wave Function Methods 386

12.5 Bond Dissociation Curves 389

12.5.1 Wave Function Methods 389

12.5.2 Density Functional Methods 394

12.6 Angle Bending Curves 394

12.7 Problematic Systems 396

12.7.1 The Geometry of FOOF 396

12.7.2 The Dipole Moment of CO 397

12.7.3 The Vibrational Frequencies of O3 398

12.8 Relative Energies of C4H6 Isomers 399

References 402

13 Optimization Techniques 404

13.1 Optimizing Quadratic Functions 405

13.2 Optimizing General Functions: Finding Minima 407

13.2.1 Steepest Descent 407

13.2.2 Conjugate Gradient Methods 408

13.2.3 Newton–Raphson Methods 409

13.2.4 Augmented Hessian Methods 410

13.2.5 Hessian Update Methods 411

13.2.6 Truncated Hessian Methods 413

13.2.7 Extrapolation: The DIIS Method 413

13.3 Choice of Coordinates 415

13.4 Optimizing General Functions: Finding Saddle Points (Transition Structures) 418

13.4.1 One-Structure Interpolation Methods 419

13.4.2 Two-Structure Interpolation Methods 421

13.4.3 Multistructure Interpolation Methods 422

13.4.4 Characteristics of Interpolation Methods 426

13.4.5 Local Methods: Gradient Norm Minimization 427

13.4.6 Local Methods: Newton–Raphson 427

13.4.7 Local Methods: The Dimer Method 429

13.4.8 Coordinates for TS Searches 429

13.4.9 Characteristics of Local Methods 430

13.4.10 Dynamic Methods 431

13.5 Constrained Optimizations 431

13.6 Global Minimizations and Sampling 433

13.6.1 Stochastic and Monte Carlo Methods 434

13.6.2 Molecular Dynamics Methods 436

13.6.3 Simulated Annealing 436

13.6.4 Genetic Algorithms 437

13.6.5 Particle Swarm and Gravitational Search Methods 437

13.6.6 Diffusion Methods 438

13.6.7 Distance Geometry Methods 439

13.6.8 Characteristics of Global Optimization Methods 439

13.7 Molecular Docking 440

13.8 Intrinsic Reaction Coordinate Methods 441

References 444

14 Statistical Mechanics and Transition State Theory 447

14.1 Transition State Theory 447

14.2 Rice–Ramsperger–Kassel–Marcus Theory 450

14.3 Dynamical Effects 451

14.4 Statistical Mechanics 452

14.5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation 454

14.5.1 Translational Degrees of Freedom 455

14.5.2 Rotational Degrees of Freedom 455

14.5.3 Vibrational Degrees of Freedom 457

14.5.4 Electronic Degrees of Freedom 458

14.5.5 Enthalpy and Entropy Contributions 459

14.6 Condensed Phases 464

References 468

15 Simulation Techniques 469

15.1 Monte Carlo Methods 472

15.1.1 Generating Non-natural Ensembles 474

15.2 Time-Dependent Methods 474

15.2.1 Molecular Dynamics Methods 474

15.2.2 Generating Non-natural Ensembles 478

15.2.3 Langevin Methods 479

15.2.4 Direct Methods 479

15.2.5 Ab Initio Molecular Dynamics 480

15.2.6 Quantum Dynamical Methods Using Potential Energy Surfaces 483

15.2.7 Reaction Path Methods 484

15.2.8 Non-Born–Oppenheimer Methods 487

15.2.9 Constrained and Biased Sampling Methods 488

15.3 Periodic Boundary Conditions 491

15.4 Extracting Information from Simulations 494

15.5 Free Energy Methods 499

15.5.1 Thermodynamic Perturbation Methods 499

15.5.2 Thermodynamic Integration Methods 500

15.6 Solvation Models 502

15.6.1 Continuum Solvation Models 503

15.6.2 Poisson–Boltzmann Methods 505

15.6.3 Born/Onsager/Kirkwood Models 506

15.6.4 Self-Consistent Reaction Field Models 508

References 511

16 Qualitative Theories 515

16.1 Frontier Molecular Orbital Theory 515

16.2 Concepts from Density Functional Theory 519

16.3 Qualitative Molecular Orbital Theory 522

16.4 Energy Decomposition Analyses 524

16.5 Orbital Correlation Diagrams: The Woodward–Hoffmann Rules 526

16.6 The Bell–Evans–Polanyi Principle/Hammond Postulate/Marcus Theory 534

16.7 More O’Ferrall–Jencks Diagrams 538

References 541

17 Mathematical Methods 543

17.1 Numbers, Vectors, Matrices and Tensors 543

17.2 Change of Coordinate System 549

17.2.1 Examples of Changing the Coordinate System 554

17.2.2 Vibrational Normal Coordinates 555

17.2.3 Energy of a Slater Determinant 557

17.2.4 Energy of a CI Wave Function 558

17.2.5 Computational Considerations 558

17.3 Coordinates, Functions, Functionals, Operators and Superoperators 560

17.3.1 Differential Operators 562

17.4 Normalization, Orthogonalization and Projection 563

17.5 Differential Equations 565

17.5.1 Simple First-Order Differential Equations 565

17.5.2 Less Simple First-Order Differential Equations 566

17.5.3 Simple Second-Order Differential Equations 566

17.5.4 Less Simple Second-Order Differential Equations 567

17.5.5 Second-Order Differential Equations Depending on the Function Itself 568

17.6 Approximating Functions 568

17.6.1 Taylor Expansion 569

17.6.2 Basis Set Expansion 570

17.6.3 Tensor Decomposition Methods 572

17.6.4 Examples of Tensor Decompositions 574

17.7 Fourier and Laplace Transformations 577

17.8 Surfaces 577

References 580

18 Statistics and QSAR 581

18.1 Introduction 581

18.2 Elementary Statistical Measures 583

18.3 Correlation between Two Sets of Data 585

18.4 Correlation between Many Sets of Data 588

18.4.1 Quality Measures 589

18.4.2 Multiple Linear Regression 590

18.4.3 Principal Component Analysis 591

18.4.4 Partial Least Squares 593

18.4.5 Illustrative Example 594

18.5 Quantitative Structure–Activity Relationships (QSAR) 595

18.6 Non-linear Correlation Methods 597

18.7 Clustering Methods 598

References 604

19 Concluding Remarks 605

Appendix A 608

Notation 608

Appendix B 614

The Variational Principle 614

The Hohenberg–Kohn Theorems 615

The Adiabatic Connection Formula 616

Reference 617

Appendix C 618

Atomic Units 618

Appendix D 619

Z Matrix Construction 619

Appendix E 627

First and Second Quantization 627

References 628

Index 629