Essentials of Computational Chemistry

Christopher Cramer, Professor of Computational Chemistry Department of Chemistry, University of Minnesota,Minneapolis, USA

Preface to the First Edition xv

Preface to the Second Edition xix

Acknowledgments xxi

1 What are Theory, Computation, and Modeling? 1

1.1 Definition of Terms 1

1.2 Quantum Mechanics 4

1.3 Computable Quantities 5

1.3.1 Structure 5

1.3.2 Potential Energy Surfaces 6

1.3.3 Chemical Properties 10

1.4 Cost and Efficiency 11

1.4.1 Intrinsic Value 11

1.4.2 Hardware and Software 12

1.4.3 Algorithms 14

1.5 Note on Units 15

Bibliography and Suggested Additional Reading 15

References 16

2 Molecular Mechanics 17

2.1 History and Fundamental Assumptions 17

2.2 Potential Energy Functional Forms 19

2.2.1 Bond Stretching 19

2.2.2 Valence Angle Bending 21

2.2.3 Torsions 22

2.2.4 van der Waals Interactions 27

2.2.5 Electrostatic Interactions 30

2.2.6 Cross Terms and Additional Non-bonded Terms 34

2.2.7 Parameterization Strategies 36

2.3 Force-field Energies and Thermodynamics 39

2.4 Geometry Optimization 40

2.4.1 Optimization Algorithms 41

2.4.2 Optimization Aspects Specific to Force Fields 46

2.5 Menagerie of Modern Force Fields 50

2.5.1 Available Force Fields 50

2.5.2 Validation 59

2.6 Force Fields and Docking 62

2.7 Case Study: (2R∗,4S∗)-1-Hydroxy-2,4-dimethylhex-5-ene 64

Bibliography and Suggested Additional Reading 66

References 67

3 Simulations of Molecular Ensembles 69

3.1 Relationship Between MM Optima and Real Systems 69

3.2 Phase Space and Trajectories 70

3.2.1 Properties as Ensemble Averages 70

3.2.2 Properties as Time Averages of Trajectories 71

3.3 Molecular Dynamics 72

3.3.1 Harmonic Oscillator Trajectories 72

3.3.2 Non-analytical Systems 74

3.3.3 Practical Issues in Propagation 77

3.3.4 Stochastic Dynamics 79

3.4 Monte Carlo 80

3.4.1 Manipulation of Phase-space Integrals 80

3.4.2 Metropolis Sampling 81

3.5 Ensemble and Dynamical Property Examples 82

3.6 Key Details in Formalism 88

3.6.1 Cutoffs and Boundary Conditions 88

3.6.2 Polarization 90

3.6.3 Control of System Variables 91

3.6.4 Simulation Convergence 93

3.6.5 The Multiple Minima Problem 96

3.7 Force Field Performance in Simulations 98

3.8 Case Study: Silica Sodalite 99

Bibliography and Suggested Additional Reading 101

References 102

4 Foundations of Molecular Orbital Theory 105

4.1 Quantum Mechanics and the Wave Function 105

4.2 The Hamiltonian Operator 106

4.2.1 General Features 106

4.2.2 The Variational Principle 108

4.2.3 The Born–Oppenheimer Approximation 110

4.3 Construction of Trial Wave Functions 111

4.3.1 The LCAO Basis Set Approach 111

4.3.2 The Secular Equation 113

4.4 H¨uckel Theory 115

4.4.1 Fundamental Principles 115

4.4.2 Application to the Allyl System 116

4.5 Many-electron Wave Functions 119

4.5.1 Hartree-product Wave Functions 120

4.5.2 The Hartree Hamiltonian 121

4.5.3 Electron Spin and Antisymmetry 122

4.5.4 Slater Determinants 124

4.5.5 The Hartree-Fock Self-consistent Field Method 126

Bibliography and Suggested Additional Reading 129

References 130

5 Semiempirical Implementations of Molecular Orbital Theory 131

5.1 Semiempirical Philosophy 131

5.1.1 Chemically Virtuous Approximations 131

5.1.2 Analytic Derivatives 133

5.2 Extended H¨uckel Theory 134

5.3 CNDO Formalism 136

5.4 INDO Formalism 139

5.4.1 INDO and INDO/S 139

5.4.2 MINDO/3 and SINDO1 141

5.5 Basic NDDO Formalism 143

5.5.1 MNDO 143

5.5.2 AM1 145

5.5.3 PM3 146

5.6 General Performance Overview of Basic NDDO Models 147

5.6.1 Energetics 147

5.6.2 Geometries 150

5.6.3 Charge Distributions 151

5.7 Ongoing Developments in Semiempirical MO Theory 152

5.7.1 Use of Semiempirical Properties in SAR 152

5.7.2 d Orbitals in NDDO Models 153

5.7.3 SRP Models 155

5.7.4 Linear Scaling 157

5.7.5 Other Changes in Functional Form 157

5.8 Case Study: Asymmetric Alkylation of Benzaldehyde 159

Bibliography and Suggested Additional Reading 162

References 163

6 Ab Initio Implementations of Hartree–Fock Molecular Orbital Theory 165

6.1 Ab Initio Philosophy 165

6.2 Basis Sets 166

6.2.1 Functional Forms 167

6.2.2 Contracted Gaussian Functions 168

6.2.3 Single-ζ , Multiple-ζ , and Split-Valence 170

6.2.4 Polarization Functions 173

6.2.5 Diffuse Functions 176

6.2.6 The HF Limit 176

6.2.7 Effective Core Potentials 178

6.2.8 Sources 180

6.3 Key Technical and Practical Points of Hartree–Fock Theory 180

6.3.1 SCF Convergence 181

6.3.2 Symmetry 182

6.3.3 Open-shell Systems 188

6.3.4 Efficiency of Implementation and Use 190

6.4 General Performance Overview of Ab Initio HF Theory 192

6.4.1 Energetics 192

6.4.2 Geometries 196

6.4.3 Charge Distributions 198

6.5 Case Study: Polymerization of 4-Substituted Aromatic Enynes 199

Bibliography and Suggested Additional Reading 201

References 201

7 Including Electron Correlation in Molecular Orbital Theory 203

7.1 Dynamical vs. Non-dynamical Electron Correlation 203

7.2 Multiconfiguration Self-Consistent Field Theory 205

7.2.1 Conceptual Basis 205

7.2.2 Active Space Specification 207

7.2.3 Full Configuration Interaction 211

7.3 Configuration Interaction 211

7.3.1 Single-determinant Reference 211

7.3.2 Multireference 216

7.4 Perturbation Theory 216

7.4.1 General Principles 216

7.4.2 Single-reference 219

7.4.3 Multireference 223

7.4.4 First-order Perturbation Theory for Some Relativistic Effects 223

7.5 Coupled-cluster Theory 224

7.6 Practical Issues in Application 227

7.6.1 Basis Set Convergence 227

7.6.2 Sensitivity to Reference Wave Function 230

7.6.3 Price/Performance Summary 235

7.7 Parameterized Methods 237

7.7.1 Scaling Correlation Energies 238

7.7.2 Extrapolation 239

7.7.3 Multilevel Methods 239

7.8 Case Study: Ethylenedione Radical Anion 244

Bibliography and Suggested Additional Reading 246

References 247

8 Density Functional Theory 249

8.1 Theoretical Motivation 249

8.1.1 Philosophy 249

8.1.2 Early Approximations 250

8.2 Rigorous Foundation 252

8.2.1 The Hohenberg–Kohn Existence Theorem 252

8.2.2 The Hohenberg–Kohn Variational Theorem 254

8.3 Kohn–Sham Self-consistent Field Methodology 255

8.4 Exchange-correlation Functionals 257

8.4.1 Local Density Approximation 258

8.4.2 Density Gradient and Kinetic Energy Density Corrections 263

8.4.3 Adiabatic Connection Methods 264

8.4.4 Semiempirical DFT 268

8.5 Advantages and Disadvantages of DFT Compared to MO Theory 271

8.5.1 Densities vs. Wave Functions 271

8.5.2 Computational Efficiency 273

8.5.3 Limitations of the KS Formalism 274

8.5.4 Systematic Improvability 278

8.5.5 Worst-case Scenarios 278

8.6 General Performance Overview of DFT 280

8.6.1 Energetics 280

8.6.2 Geometries 291

8.6.3 Charge Distributions 294

8.7 Case Study: Transition-Metal Catalyzed Carbonylation of Methanol 299

Bibliography and Suggested Additional Reading 300

References 301

9 Charge Distribution and Spectroscopic Properties 305

9.1 Properties Related to Charge Distribution 305

9.1.1 Electric Multipole Moments 305

9.1.2 Molecular Electrostatic Potential 308

9.1.3 Partial Atomic Charges 309

9.1.4 Total Spin 324

9.1.5 Polarizability and Hyperpolarizability 325

9.1.6 ESR Hyperfine Coupling Constants 327

9.2 Ionization Potentials and Electron Affinities 330

9.3 Spectroscopy of Nuclear Motion 331

9.3.1 Rotational 332

9.3.2 Vibrational 334

9.4 NMR Spectral Properties 344

9.4.1 Technical Issues 344

9.4.2 Chemical Shifts and Spin–spin Coupling Constants 345

9.5 Case Study: Matrix Isolation of Perfluorinated p-Benzyne 349

Bibliography and Suggested Additional Reading 351

References 351

10 Thermodynamic Properties 355

10.1 Microscopic–macroscopic Connection 355

10.2 Zero-point Vibrational Energy 356

10.3 Ensemble Properties and Basic Statistical Mechanics 357

10.3.1 Ideal Gas Assumption 358

10.3.2 Separability of Energy Components 359

10.3.3 Molecular Electronic Partition Function 360

10.3.4 Molecular Translational Partition Function 361

10.3.5 Molecular Rotational Partition Function 362

10.3.6 Molecular Vibrational Partition Function 364

10.4 Standard-state Heats and Free Energies of Formation and Reaction 366

10.4.1 Direct Computation 367

10.4.2 Parametric Improvement 370

10.4.3 Isodesmic Equations 372

10.5 Technical Caveats 375

10.5.1 Semiempirical Heats of Formation 375

10.5.2 Low-frequency Motions 375

10.5.3 Equilibrium Populations over Multiple Minima 377

10.5.4 Standard-state Conversions 378

10.5.5 Standard-state Free Energies, Equilibrium Constants, and Concentrations 379

10.6 Case Study: Heat of Formation of H2NOH 381

Bibliography and Suggested Additional Reading 383

References 383

11 Implicit Models for Condensed Phases 385

11.1 Condensed-phase Effects on Structure and Reactivity 385

11.1.1 Free Energy of Transfer and Its Physical Components 386

11.1.2 Solvation as It Affects Potential Energy Surfaces 389

11.2 Electrostatic Interactions with a Continuum 393

11.2.1 The Poisson Equation 394

11.2.2 Generalized Born 402

11.2.3 Conductor-like Screening Model 404

11.3 Continuum Models for Non-electrostatic Interactions 406

11.3.1 Specific Component Models 406

11.3.2 Atomic Surface Tensions 407

11.4 Strengths and Weaknesses of Continuum Solvation Models 410

11.4.1 General Performance for Solvation Free Energies 410

11.4.2 Partitioning 416

11.4.3 Non-isotropic Media 416

11.4.4 Potentials of Mean Force and Solvent Structure 419

11.4.5 Molecular Dynamics with Implicit Solvent 420

11.4.6 Equilibrium vs. Non-equilibrium Solvation 421

11.5 Case Study: Aqueous Reductive Dechlorination of Hexachloroethane 422

Bibliography and Suggested Additional Reading 424

References 425

12 Explicit Models for Condensed Phases 429

12.1 Motivation 429

12.2 Computing Free-energy Differences 429

12.2.1 Raw Differences 430

12.2.2 Free-energy Perturbation 432

12.2.3 Slow Growth and Thermodynamic Integration 435

12.2.4 Free-energy Cycles 437

12.2.5 Potentials of Mean Force 439

12.2.6 Technical Issues and Error Analysis 443

12.3 Other Thermodynamic Properties 444

12.4 Solvent Models 445

12.4.1 Classical Models 445

12.4.2 Quantal Models 447

12.5 Relative Merits of Explicit and Implicit Solvent Models 448

12.5.1 Analysis of Solvation Shell Structure and Energetics 448

12.5.2 Speed/Efficiency 450

12.5.3 Non-equilibrium Solvation 450

12.5.4 Mixed Explicit/Implicit Models 451

12.6 Case Study: Binding of Biotin Analogs to Avidin 452

Bibliography and Suggested Additional Reading 454

References 455

13 Hybrid Quantal/Classical Models 457

13.1 Motivation 457

13.2 Boundaries Through Space 458

13.2.1 Unpolarized Interactions 459

13.2.2 Polarized QM/Unpolarized MM 461

13.2.3 Fully Polarized Interactions 466

13.3 Boundaries Through Bonds 467

13.3.1 Linear Combinations of Model Compounds 467

13.3.2 Link Atoms 473

13.3.3 Frozen Orbitals 475

13.4 Empirical Valence Bond Methods 477

13.4.1 Potential Energy Surfaces 478

13.4.2 Following Reaction Paths 480

13.4.3 Generalization to QM/MM 481

13.5 Case Study: Catalytic Mechanism of Yeast Enolase 482

Bibliography and Suggested Additional Reading 484

References 485

14 Excited Electronic States 487

14.1 Determinantal/Configurational Representation of Excited States 487

14.2 Singly Excited States 492

14.2.1 SCF Applicability 493

14.2.2 CI Singles 496

14.2.3 Rydberg States 498

14.3 General Excited State Methods 499

14.3.1 Higher Roots in MCSCF and CI Calculations 499

14.3.2 Propagator Methods and Time-dependent DFT 501

14.4 Sum and Projection Methods 504

14.5 Transition Probabilities 507

14.6 Solvatochromism 511

14.7 Case Study: Organic Light Emitting Diode Alq3 513

Bibliography and Suggested Additional Reading 515

References 516

15 Adiabatic Reaction Dynamics 519

15.1 Reaction Kinetics and Rate Constants 519

15.1.1 Unimolecular Reactions 520

15.1.2 Bimolecular Reactions 521

15.2 Reaction Paths and Transition States 522

15.3 Transition-state Theory 524

15.3.1 Canonical Equation 524

15.3.2 Variational Transition-state Theory 531

15.3.3 Quantum Effects on the Rate Constant 533

15.4 Condensed-phase Dynamics 538

15.5 Non-adiabatic Dynamics 539

15.5.1 General Surface Crossings 539

15.5.2 Marcus Theory 541

15.6 Case Study: Isomerization of Propylene Oxide 544

Bibliography and Suggested Additional Reading 546

References 546

Appendix A Acronym Glossary 549

Appendix B Symmetry and Group Theory 557

B.1 Symmetry Elements 557

B.2 Molecular Point Groups and Irreducible Representations 559

B.3 Assigning Electronic State Symmetries 561

B.4 Symmetry in the Evaluation of Integrals and Partition Functions 562

Appendix C Spin Algebra 565

C.1 Spin Operators 565

C.2 Pure- and Mixed-spin Wave Functions 566

C.3 UHF Wave Functions 571

C.4 Spin Projection/Annihilation 571

Reference 574

Appendix D Orbital Localization 575

D.1 Orbitals as Empirical Constructs 575

D.2 Natural Bond Orbital Analysis 578

References 579

Index 581