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Introduction to Computational Chemistry
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Table of Contents

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 262 6.9 Time-Dependent Density Functional Theory (TDDFT) 263 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 7.5 Huckel Theory 283 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 Schroedinger Equations 302 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 10.4 Localized Orbitals 329 10.5 Natural Orbitals 333 10.6 Computational Considerations 337 10.7 Examples 338 References 339 11 Molecular Properties 341 11.1 Examples of Molecular Properties 343 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 11.7 Electric Field Perturbation 357 11.8 Magnetic Field Perturbation 358 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 12.2 Total Energy Convergence 383 12.3 Dipole Moment Convergence 385 12.4 Vibrational Frequency Convergence 386 12.5 Bond Dissociation Curves 389 12.6 Angle Bending Curves 394 12.7 Problematic Systems 396 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 13.3 Choice of Coordinates 415 13.4 Optimizing General Functions: Finding Saddle Points (Transition Structures) 418 13.5 Constrained Optimizations 431 13.6 Global Minimizations and Sampling 433 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 StatisticalMechanics 452 14.5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation 454 14.6 Condensed Phases 464 References 468 15 Simulation Techniques 469 15.1 Monte Carlo Methods 472 15.2 Time-Dependent Methods 474 15.3 Periodic Boundary Conditions 491 15.4 Extracting Information from Simulations 494 15.5 Free Energy Methods 499 15.6 Solvation Models 502 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: TheWoodward-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 17.3 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 17.6 Approximating Functions 568 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 18.5 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

About the Author

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).

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