Frank Jensen
Introduction to Computational Chemistry (eBook, PDF)
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Frank Jensen
Introduction to Computational Chemistry (eBook, PDF)
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Introduction to Computational Chemistry 3rd Edition provides a comprehensive account of the fundamental principles underlying different computational methods. Fully revised and updated throughout to reflect important method developments and improvements since publication of the previous edition, this timely update includes the following significant revisions and new topics:
Polarizable force fields | Tight-binding DFT | More extensive DFT functionals, excited states and time dependent molecular properties | Accelerated Molecular Dynamics methods | Tensor decomposition methods | Cluster…mehr
- Geräte: PC
- eBook Hilfe
Introduction to Computational Chemistry 3rd Edition provides a comprehensive account of the fundamental principles underlying different computational methods. Fully revised and updated throughout to reflect important method developments and improvements since publication of the previous edition, this timely update includes the following significant revisions and new topics:
Additional information is available at: www.wiley.com/go/jensen/computationalchemistry3
- Polarizable force fields
- Tight-binding DFT
- More extensive DFT functionals, excited states and time dependent molecular properties
- Accelerated Molecular Dynamics methods
- Tensor decomposition methods
- Cluster analysis
- Reduced scaling and reduced prefactor methods
Additional information is available at: www.wiley.com/go/jensen/computationalchemistry3
Produktdetails
- Produktdetails
- Verlag: John Wiley & Sons
- Erscheinungstermin: 28. November 2016
- Englisch
- ISBN-13: 9781118825983
- Artikelnr.: 47381678
- Verlag: John Wiley & Sons
- Erscheinungstermin: 28. November 2016
- Englisch
- ISBN-13: 9781118825983
- Artikelnr.: 47381678
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).
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
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
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
1.7 Classical Mechanics 11
1.8 Quantum Mechanics 13
1.9 Chemistry 18
References 19
2 Force Field Methods 20
2.1 Introduction 20
2.2 The Force Field Energy 21
2.3 Force Field Parameterization 53
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
2.12 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-FockTheory 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
3.9 Periodic Systems 119
References 121
4 Electron Correlation Methods 124
4.1 Excited Slater Determinants 125
4.2 Configuration Interaction 128
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
4.9 Coupled Cluster 157
4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 162
4.11 Methods Involving the Interelectronic Distance 166
4.12 Techniques for Improving the Computational Efficiency 169
4.13 Summary of Electron Correlation Methods 174
4.14 Excited States 176
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
5.4 Examples of Standard Basis Sets 200
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
5.11 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
6.6 Performance of Density Functional Methods 258
6.7 Computational Considerations 260
6.8 Differences between Density Functional Theory and Hartree-Fock
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
1.7 Classical Mechanics 11
1.8 Quantum Mechanics 13
1.9 Chemistry 18
References 19
2 Force Field Methods 20
2.1 Introduction 20
2.2 The Force Field Energy 21
2.3 Force Field Parameterization 53
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
2.12 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-FockTheory 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
3.9 Periodic Systems 119
References 121
4 Electron Correlation Methods 124
4.1 Excited Slater Determinants 125
4.2 Configuration Interaction 128
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
4.9 Coupled Cluster 157
4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 162
4.11 Methods Involving the Interelectronic Distance 166
4.12 Techniques for Improving the Computational Efficiency 169
4.13 Summary of Electron Correlation Methods 174
4.14 Excited States 176
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
5.4 Examples of Standard Basis Sets 200
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
5.11 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
6.6 Performance of Density Functional Methods 258
6.7 Computational Considerations 260
6.8 Differences between Density Functional Theory and Hartree-Fock
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
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
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
1.7 Classical Mechanics 11
1.8 Quantum Mechanics 13
1.9 Chemistry 18
References 19
2 Force Field Methods 20
2.1 Introduction 20
2.2 The Force Field Energy 21
2.3 Force Field Parameterization 53
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
2.12 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-FockTheory 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
3.9 Periodic Systems 119
References 121
4 Electron Correlation Methods 124
4.1 Excited Slater Determinants 125
4.2 Configuration Interaction 128
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
4.9 Coupled Cluster 157
4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 162
4.11 Methods Involving the Interelectronic Distance 166
4.12 Techniques for Improving the Computational Efficiency 169
4.13 Summary of Electron Correlation Methods 174
4.14 Excited States 176
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
5.4 Examples of Standard Basis Sets 200
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
5.11 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
6.6 Performance of Density Functional Methods 258
6.7 Computational Considerations 260
6.8 Differences between Density Functional Theory and Hartree-Fock
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
1.7 Classical Mechanics 11
1.8 Quantum Mechanics 13
1.9 Chemistry 18
References 19
2 Force Field Methods 20
2.1 Introduction 20
2.2 The Force Field Energy 21
2.3 Force Field Parameterization 53
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
2.12 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-FockTheory 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
3.9 Periodic Systems 119
References 121
4 Electron Correlation Methods 124
4.1 Excited Slater Determinants 125
4.2 Configuration Interaction 128
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
4.9 Coupled Cluster 157
4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 162
4.11 Methods Involving the Interelectronic Distance 166
4.12 Techniques for Improving the Computational Efficiency 169
4.13 Summary of Electron Correlation Methods 174
4.14 Excited States 176
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
5.4 Examples of Standard Basis Sets 200
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
5.11 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
6.6 Performance of Density Functional Methods 258
6.7 Computational Considerations 260
6.8 Differences between Density Functional Theory and Hartree-Fock