Introduction to Computational Chemistry

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