Introduction to Computational Chemistry

Second Edition

Introduction toComputational Chemistry

Second Edition

Frank JensenDepartment of Chemistry, University of Southern Denmark, Odense, Denmark

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Jensen, Frank.Introduction to computational chemistry / Frank Jensen. – 2nd ed.

p. cm.Includes bibliographical references and index.ISBN-13: 978-0-470-01186-7 (cloth : alk. paper)ISBN-10: 0-470-01186-6 (cloth : alk. paper)ISBN-13: 978-0-470-01187-4 (pbk. : alk. paper)ISBN-10: 0-470-01187-4 (pbk. : alk. paper)1. Chemistry, Physical and theoretical – Data processing. 2. Chemistry, Physical and

theoretical – Mathematics. I. Title.QD455.3.E4J46 2006541.0285 – dc22

2006023998

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Contents

Preface to the First Edition xvPreface to the Second Edition xix

1 Introduction 1

1.1 Fundamental Issues 21.2 Describing the System 31.3 Fundamental Forces 41.4 The Dynamical Equation 51.5 Solving the Dynamical Equation 81.6 Separation of Variables 8

1.6.1 Separating space and time variables 101.6.2 Separating nuclear and electronic variables 101.6.3 Separating variables in general 11

1.7 Classical Mechanics 121.7.1 The Sun–Earth system 121.7.2 The solar system 13

1.8 Quantum Mechanics 141.8.1 A hydrogen-like atom 141.8.2 The helium atom 17

1.9 Chemistry 19References 21

2 Force Field Methods 22

2.1 Introduction 222.2 The Force Field Energy 24

2.2.1 The stretch energy 252.2.2 The bending energy 272.2.3 The out-of-plane bending energy 302.2.4 The torsional energy 302.2.5 The van der Waals energy 342.2.6 The electrostatic energy: charges and dipoles 402.2.7 The electrostatic energy: multipoles and polarizabilities 43

2.2.8 Cross terms 472.2.9 Small rings and conjugated systems 482.2.10 Comparing energies of structurally different molecules 50

2.3 Force Field Parameterization 512.3.1 Parameter reductions in force fields 572.3.2 Force fields for metal coordination compounds 582.3.3 Universal force fields 62

2.4 Differences in Force Fields 622.5 Computational Considerations 652.6 Validation of Force Fields 672.7 Practical Considerations 692.8 Advantages and Limitations of Force Field Methods 692.9 Transition Structure Modelling 70

2.9.1 Modelling the TS as a minimum energy structure 702.9.2 Modelling the TS as a minimum energy structure on the reactant/

product energy seam 712.9.3 Modelling the reactive energy surface by interacting force

field functions or by geometry-dependent parameters 732.10 Hybrid Force Field Electronic Structure Methods 74

References 77

3 Electronic Structure Methods: Independent-Particle Models 80

3.1 The Adiabatic and Born–Oppenheimer Approximations 823.2 Self-Consistent Field Theory 863.3 The Energy of a Slater Determinant 873.4 Koopmans’ Theorem 923.5 The Basis Set Approximation 933.6 An Alternative Formulation of the Variational Problem 983.7 Restricted and Unrestricted Hartree–Fock 993.8 SCF Techniques 100

3.8.1 SCF convergence 1013.8.2 Use of symmetry 1043.8.3 Ensuring that the HF energy is a minimum, and the

correct minimum 1053.8.4 Initial guess orbitals 1073.8.5 Direct SCF 1083.8.6 Reduced scaling techniques 110

3.9 Periodic Systems 1133.10 Semi-Empirical Methods 115

3.10.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 1163.10.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 1173.10.3 Complete Neglect of Differential Overlap Approximation (CNDO) 117

3.11 Parameterization 1183.11.1 Modified Intermediate Neglect of Differential Overlap (MINDO) 1193.11.2 Modified NDDO models 1193.11.3 Modified Neglect of Diatomic Overlap (MNDO) 1213.11.4 Austin Model 1 (AM1) 1213.11.5 Modified Neglect of Diatomic Overlap, Parametric Method Number 3 (PM3) 1223.11.6 Parametric Method number 5 (PM5) and PDDG/PM3 methods 123

vi CONTENTS

3.11.7 The MNDO/d and AM1/d methods 1243.11.8 Semi Ab initio Method 1 124

3.12 Performance of Semi-Empirical Methods 1253.13 Hückel Theory 127

3.13.1 Extended Hückel theory 1273.13.2 Simple Hückel theory 128

3.14 Limitations and Advantages of Semi-Empirical Methods 129References 131

4 Electron Correlation Methods 133

4.1 Excited Slater Determinants 1354.2 Configuration Interaction 137

4.2.1 CI Matrix elements 1384.2.2 Size of the CI matrix 1414.2.3 Truncated CI methods 1434.2.4 Direct CI methods 144

4.3 Illustrating how CI Accounts for Electron Correlation, and theRHF Dissociation Problem 145

4.4 The UHF Dissociation, and the Spin Contamination Problem 1484.5 Size Consistency and Size Extensivity 1534.6 Multi-Configuration Self-Consistent Field 1534.7 Multi-Reference Configuration Interaction 1584.8 Many-Body Perturbation Theory 159

4.8.1 Møller–Plesset perturbation theory 1624.8.2 Unrestricted and projected Møller–Plesset methods 168

4.9 Coupled Cluster 1694.9.1 Truncated coupled cluster methods 172

4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory 1744.10.1 Illustrating correlation methods for the beryllium atom 177

4.11 Methods Involving the Interelectronic Distance 1784.12 Direct Methods 1814.13 Localized Orbital Methods 1824.14 Summary of Electron Correlation Methods 1834.15 Excited States 1864.16 Quantum Monte Carlo Methods 187

References 189

5 Basis Sets 192

5.1 Slater and Gaussian Type Orbitals 1925.2 Classification of Basis Sets 1945.3 Even- and Well-Tempered Basis Sets 1985.4 Contracted Basis Sets 200

5.4.1 Pople style basis sets 2025.4.2 Dunning–Huzinaga basis sets 2045.4.3 MINI, MIDI and MAXI basis sets 2055.4.4 Ahlrichs type basis sets 2055.4.5 Atomic natural orbital basis sets 2055.4.6 Correlation consistent basis sets 206

CONTENTS vii

5.4.7 Polarization consistent basis sets 2075.4.8 Basis set extrapolation 208

5.5 Plane Wave Basis Functions 2115.6 Recent Developments and Computational Issues 2125.7 Composite Extrapolation Procedures 2135.8 Isogyric and Isodesmic Reactions 2215.9 Effective Core Potentials 2225.10 Basis Set Superposition Errors 2255.11 Pseudospectral Methods 227

References 229

6 Density Functional Methods 232

6.1 Orbital-Free Density Functional Theory 2336.2 Kohn–Sham Theory 2356.3 Reduced Density Matrix Methods 2366.4 Exchange and Correlation Holes 2406.5 Exchange–Correlation Functionals 243

6.5.1 Local Density Approximation 2466.5.2 Gradient-corrected methods 2486.5.3 Higher order gradient or meta-GGA methods 2506.5.4 Hybrid or hyper-GGA methods 2526.5.5 Generalized random phase methods 2536.5.6 Functionals overview 254

6.6 Performance and Properties of Density Functional Methods 2556.7 DFT Problems 2586.8 Computational Considerations 2606.9 Final Considerations 263

References 264

7 Valence Bond Methods 268

7.1 Classical Valence Bond Theory 2697.2 Spin-Coupled Valence Bond Theory 2707.3 Generalized Valence Bond Theory 275

References 276

8 Relativistic Methods 277

8.1 The Dirac Equation 2788.2 Connections Between the Dirac and Schrödinger Equations 280

8.2.1 Including electric potentials 2808.2.2 Including both electric and magnetic potentials 282

8.3 Many-Particle Systems 2848.4 Four-Component Calculations 2878.5 Relativistic Effects 289

References 292

9 Wave Function Analysis 293

9.1 Population Analysis Based on Basis Functions 2939.2 Population Analysis Based on the Electrostatic Potential 296

viii CONTENTS

9.3 Population Analysis Based on the Electron Density 2999.3.1 Atoms In Molecules 2999.3.2 Voronoi, Hirshfeld and Stewart atomic charges 3039.3.3 Generalized atomic polar tensor charges 304

9.4 Localized Orbitals 3049.4.1 Computational considerations 306

9.5 Natural Orbitals 3089.6 Natural Atomic Orbital and Natural Bond Orbital Analysis 3099.7 Computational Considerations 3119.8 Examples 312

References 313

10 Molecular Properties 315

10.1 Examples of Molecular Properties 31610.1.1 External electric field 31610.1.2 External magnetic field 31810.1.3 Internal magnetic moments 31810.1.4 Geometry change 31910.1.5 Mixed derivatives 319

10.2 Perturbation Methods 32110.3 Derivative Techniques 32110.4 Lagrangian Techniques 32410.5 Coupled Perturbed Hartree–Fock 32510.6 Electric Field Perturbation 329

10.6.1 External electric field 32910.6.2 Internal electric field 329

10.7 Magnetic Field Perturbation 32910.7.1 External magnetic field 33110.7.2 Nuclear spin 33210.7.3 Electron spin 33310.7.4 Classical terms 33310.7.5 Relativistic terms 33410.7.6 Magnetic properties 33410.7.7 Gauge dependence of magnetic properties 338

10.8 Geometry Perturbations 33910.9 Response and Propagator Methods 34310.10 Property Basis Sets 348

References 349

11 Illustrating the Concepts 350

11.1 Geometry Convergence 35011.1.1 Ab Initio methods 35011.1.2 Density functional methods 353

11.2 Total Energy Convergence 35411.3 Dipole Moment Convergence 356

11.3.1 Ab Initio methods 35611.3.2 Density functional methods 357

11.4 Vibrational Frequency Convergence 35811.4.1 Ab Initio methods 35811.4.2 Density functional methods 360

CONTENTS ix

11.5 Bond Dissociation Curves 36111.5.1 Basis set effect at the Hartree–Fock level 36111.5.2 Performance of different types of wave function 36311.5.3 Density functional methods 369

11.6 Angle Bending Curves 37011.7 Problematic Systems 370

11.7.1 The geometry of FOOF 37111.7.2 The dipole moment of CO 37211.7.3 The vibrational frequencies of O3 373

11.8 Relative Energies of C4H6 Isomers 374References 378

12 Optimization Techniques 380

12.1 Optimizing Quadratic Functions 38112.2 Optimizing General Functions: Finding Minima 383

12.2.1 Steepest descent 38312.2.2 Conjugate gradient methods 38412.2.3 Newton–Raphson methods 38512.2.4 Step control 38612.2.5 Obtaining the Hessian 38712.2.6 Storing and diagonalizing the Hessian 38812.2.7 Extrapolations: the GDIIS method 389

12.3 Choice of Coordinates 39012.4 Optimizing General Functions: Finding Saddle Points (Transition Structures) 394

12.4.1 One-structure interpolation methods: coordinate driving, linear and quadratic synchronous transit, and sphere optimization 394

12.4.2 Two-structure interpolation methods: saddle, line-then-plane, ridge and step-and-slide optimizations 397

12.4.3 Multi-structure interpolation methods: chain, locally updated planes, self-penalty walk, conjugate peak refinement and nudged elastic band 398

12.4.4 Characteristics of interpolation methods 40112.4.5 Local methods: gradient norm minimization 40212.4.6 Local methods: Newton–Raphson 40312.4.7 Local methods: the dimer method 40512.4.8 Coordinates for TS searches 40512.4.9 Characteristics of local methods 40612.4.10 Dynamic methods 406

12.5 Constrained Optimization Problems 40712.6 Conformational Sampling and the Global Minimum Problem 409

12.6.1 Stochastic and Monte Carlo methods 41112.6.2 Molecular dynamics 41212.6.3 Simulated annealing 41312.6.4 Genetic algorithms 41312.6.5 Diffusion methods 41412.6.6 Distance geometry methods 414

12.7 Molecular Docking 41512.8 Intrinsic Reaction Coordinate Methods 416

References 419

x CONTENTS

13 Statistical Mechanics and Transition State Theory 421

13.1 Transition State Theory 42113.2 Rice–Ramsperger–Kassel–Marcus Theory 42413.3 Dynamical Effects 42513.4 Statistical Mechanics 42613.5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation 429

13.5.1 Translational degrees of freedom 43013.5.2 Rotational degrees of freedom 43013.5.3 Vibrational degrees of freedom 43113.5.4 Electronic degrees of freedom 43313.5.5 Enthalpy and entropy contributions 433

13.6 Condensed Phases 439References 443

14 Simulation Techniques 445

14.1 Monte Carlo Methods 44814.1.1 Generating non-natural ensembles 450

14.2 Time-Dependent Methods 45014.2.1 Molecular dynamics methods 45114.2.2 Generating non-natural ensembles 45414.2.3 Langevin methods 45514.2.4 Direct methods 45514.2.5 Extended Lagrange techniques (Car–Parrinello methods) 45714.2.6 Quantum methods using potential energy surfaces 45914.2.7 Reaction path methods 46014.2.8 Non-Born–Oppenheimer methods 46314.2.9 Constrained sampling methods 463

14.3 Periodic Boundary Conditions 46414.4 Extracting Information from Simulations 46814.5 Free Energy Methods 472

14.5.1 Thermodynamic perturbation methods 47214.5.2 Thermodynamic integration methods 473

14.6 Solvation Models 47514.7 Continuum Solvation Models 476

14.7.1 Poisson–Boltzmann methods 47814.7.2 Born/Onsager/Kirkwood models 48014.7.3 Self-consistent reaction field models 481References 484

15 Qualitative Theories 487

15.1 Frontier Molecular Orbital Theory 48715.2 Concepts from Density Functional Theory 49215.3 Qualitative Molecular Orbital Theory 49415.4 Woodward–Hoffmann Rules 49715.5 The Bell–Evans–Polanyi Principle/Hammond Postulate/Marcus Theory 50615.6 More O’Ferrall–Jencks Diagrams 510

References 512

CONTENTS xi

16 Mathematical Methods 514

16.1 Numbers, Vectors, Matrices and Tensors 51416.2 Change of Coordinate System 520

16.2.1 Examples of changing the coordinate system 52516.2.2 Vibrational normal coordinates 52616.2.3 Energy of a Slater determinant 52816.2.4 Energy of a CI wave function 52916.2.5 Computational Consideration 529

16.3 Coordinates, Functions, Functionals, Operators and Superoperators 53016.3.1 Differential operators 531

16.4 Normalization, Orthogonalization and Projection 53216.5 Differential Equations 535

16.5.1 Simple first-order differential equations 53516.5.2 Less simple first-order differential equations 53616.5.3 Simple second-order differential equations 53616.5.4 Less simple second-order differential equations 53716.5.5 Second-order differential equations depending on the

function itself 53716.6 Approximating Functions 538

16.6.1 Taylor expansion 53916.6.2 Basis set expansion 541

16.7 Fourier and Laplace Transformations 54116.8 Surfaces 543

References 546

17 Statistics and QSAR 547

17.1 Introduction 54717.2 Elementary Statistical Measures 54917.3 Correlation Between Two Sets of Data 55017.4 Correlation between Many Sets of Data 553

17.4.1 Multiple-descriptor data sets and quality analysis 55317.4.2 Multiple linear regression 55517.4.3 Principal component and partial least squares analysis 55617.4.4 Illustrative example 558

17.5 Quantitative Structure–Activity Relationships (QSAR) 559References 561

18 Concluding Remarks 562

Appendix A 565Notation 565

Appendix B 570B.1 The Variational Principle 570B.2 The Hohenberg–Kohn Theorems 571B.3 The Adiabatic Connection Formula 572Reference 573

xii CONTENTS

Appendix C 574Atomic Units 574

Appendix D 575Z-Matrix Construction 575

Index 583

CONTENTS xiii

Preface to the First Edition

Computational chemistry is rapidly emerging as a subfield of theoretical chemistry,where the primary focus is on solving chemically related problems by calculations. Forthe newcomer to the field, there are three main problems:

(1) Deciphering the code. The language of computational chemistry is littered withacronyms, what do these abbreviations stand for in terms of underlying assump-tions and approximations?

(2) Technical problems. How does one actually run the program and what to look forin the output?

(3) Quality assessment. How good is the number that has been calculated?

Point (1) is part of every new field: there is not much to do about it. If you want to livein another country, you have to learn the language. If you want to use computationalchemistry methods, you need to learn the acronyms. I have tried in the present bookto include a good fraction of the most commonly used abbreviations and standard pro-cedures.

Point (2) is both hardware and software specific. It is not well suited for a text book,as the information rapidly becomes out of date.The average lifetime of computer hard-ware is a few years, the time between new versions of software is even less. Problemsof type (2) need to be solved “on location”. I have made one exception, however, andhave including a short discussion of how to make Z-matrices. A Z-matrix is a conven-ient way of specifying a molecular geometry in terms of internal coordinates, and it isused by many electronic structure programs. Furthermore, geometry optimizations areoften performed in Z-matrix variables, and since optimizations in a good set of inter-nal coordinates are significantly faster than in Cartesian coordinates, it is important tohave a reasonable understanding of Z-matrix construction.

As computer programs evolve they become easier to use. Modern programs oftencommunicate with the user in terms of a graphical interface, and many methods havebecome essential “black box” procedures: if you can draw the molecule, you can alsodo the calculation.This effectively means that you no longer have to be a highly trainedtheoretician to run even quite sophisticated calculations.

The ease with which calculations can be performed means that point (3) has becomethe central theme in computational chemistry. It is quite easy to run a series of calcu-lations which produce results that are absolutely meaningless. The program will nottell you whether the chosen method is valid for the problem you are studying. Qualityassessment is thus an absolute requirement. This, however, requires much more expe-rience and insight than just running the program. A basic understanding of the theorybehind the method is needed, and a knowledge of the performance of the method forother systems. If you are breaking new ground, where there is no previous experience,you need a way of calibrating the results.

The lack of quality assessment is probably one of the reasons why computationalchemistry has (had) a somewhat bleak reputation. “If five different computationalmethods give five widely different results, what has computational chemistry con-tributed? You just pick the number closest to experiments and claim that you canreproduce experimental data accurately.” One commonly sees statements of the type“The theoretical results for property X are in disagreement. Calculation at theCCSD(T)/6-31G(d,p) level predicts that . . . , while the MINDO/3 method gives oppos-ing results. There is thus no clear consent from theory.” This is clearly a lack of under-standing of the quality of the calculations. If the results disagree, there is a very highprobability that the CCSD(T) results are basically correct, and the MINDO/3 resultsare wrong. If you want to make predictions, and not merely reproduce known results,you need to be able to judge the quality of your results. This is by far the most diffi-cult task in computational chemistry. I hope the present book will give some idea ofthe limitations of different methods.

Computers don’t solve problems, people do. Computers just generate numbers.Although computational chemistry has evolved to the stage where it often can be com-petitive with experimental methods for generating a value for a given property of agiven molecule, the number of possible molecules (there are an estimated 10200 mole-cules with a molecular weight less than 850) and their associated properties is so hugethat only a very tiny fraction will ever be amenable to calculations (or experiments).Furthermore, with the constant increase in computational power, a calculation thatbarely can be done today will be possible on medium-sized machines in 5–10 years.Prediction of properties with methods that do not provide converged results (withrespect to theoretical level) will typically only have a lifetime of a few years beforebeing surpassed by more accurate calculations.

The real strength of computational chemistry is the ability to generate data (forexample by analyzing the wave function) from which a human may gain insight, andthereby rationalize the behaviour of a large class of molecules. Such insights and ration-alizations are much more likely to be useful over a longer period of time than the rawresults themselves. A good example is the concept used by organic chemists with mol-ecules composed of functional groups, and representing reactions by “pushing elec-trons”. This may not be particular accurate from a quantum mechanical point of view,but it is very effective in rationalizing a large body of experimental results, and hasgood predictive power.

Just as computers do not solve problems, mathematics by itself does not provideinsight. It merely provides formulas, a framework for organizing thoughts. It is in thisspirit that I have tried to write this book. Only the necessary (obviously a subjectivecriterion) mathematical background has been provided, the aim being that the reader

xvi PREFACE TO THE FIRST EDITION

should be able to understand the premises and limitations of different methods, andfollow the main steps in running a calculation. This means that I in many cases haveomitted to tell the reader of some of the finer details, which may annoy the purists.However, I believe the large overview is necessary before embarking on a more strin-gent and detailed derivation of the mathematics. The goal of this book is to providean overview of commonly used methods, giving enough theoretical background tounderstand why for example the AMBER force field is used for modelling proteinsbut MM2 is used for small organic molecules. Or why coupled cluster inherently is aniterative method, while perturbation theory and configuration interaction inherentlyare non-iterative methods, although the CI problem in practice is solved by iterativetechniques.

The prime focus of this book is on calculating molecular structures and (relative)energies, and less on molecular properties or dynamical aspects. In my experience, pre-dicting structures and energetics are the main uses of computational chemistry today,although this may well change in the coming years. I have tried to include mostmethods that are already extensively used, together with some that I expect to becomegenerally available in the near future. How detailed the methods are described dependspartly on how practical and commonly used the methods are (both in terms of com-putational resources and software), and partly reflects my own limitations in terms ofunderstanding. Although simulations (e.g. molecular dynamics) are becoming increas-ingly powerful tools, only a very rudimentary introduction is provided in Chapter 16.The area is outside my expertise, and several excellent textbooks are already available.

Computational chemistry contains a strong practical element. Theoretical methodsmust be translated into working computer programs in order to produce results. Dif-ferent algorithms, however, may have different behaviours in practice, and it becomesnecessary to be able to evaluate whether a certain type of calculation can be carriedout with the available computers. The book thus contains some guidelines for evalu-ating what type of resources necessary for carrying out a given calculation.

The present book grew out of a series of lecture notes that I have used for teachinga course in computational chemistry at Odense University, and the style of the bookreflects its origin. It is difficult to master all disciplines in the vast field of computa-tional chemistry. A special thanks to H. J. Aa. Jensen, K. V. Mikkelsen, T. Saue, S. P. A.Sauer, M. Schmidt, P. M. W. Gill, P.-O. Norrby, D. L. Cooper, T. U. Helgaker and H. G.Petersen for having read various parts of the book and providing input. Remainingerrors are of course my sole responsibility.A good part of the final transformation froma set of lecture notes to the present book was done during a sabbatical leave spentwith Prof. L. Radom at the Research School of Chemistry, Australia National Univer-sity, Canberra, Australia. A special thanks to him for his hospitality during the stay.

A few comments on the layout of the book. Definitions, acronyms or commonphrases are marked in italic; these can be found in the index. Underline is used foremphasizing important points. Operators, vectors and matrices are denoted in bold,scalars in normal text. Although I have tried to keep the notation as consistent as pos-sible, different branches in computational chemistry often use different symbols forthe same quantity. In order to comply with common usage, I have elected sometimesto switch notation between chapters. The second derivative of the energy, for example,is called the force constant k in force field theory, the corresponding matrix is denotedF when discussing vibrations, and called the Hessian H for optimization purposes.

PREFACE TO THE FIRST EDITION xvii

I have assumed that the reader has no prior knowledge of concepts specific to com-putational chemistry, but has a working understanding of introductory quantummechanics and elementary mathematics, especially linear algebra, vector, differentialand integral calculus. The following features specific to chemistry are used in thepresent book without further introduction. Adequate descriptions may be found in anumber of quantum chemistry textbooks (J. P. Lowe, Quantum Chemistry, AcademicPress, 1993; I. N. Levine, Quantum Chemistry, Prentice Hall, 1992; P. W. Atkins, Mole-cular Quantum Mechanics, Oxford University Press, 1983).

(1) The Schrödinger equation, with the consequences of quantized solutions andquantum numbers.

(2) The interpretation of the square of the wave function as a probability distribution,the Heisenberg uncertainty principle and the possibility of tunnelling.

(3) The solutions for the hydrogen atom, atomic orbitals.(4) The solutions for the harmonic oscillator and rigid rotor.(5) The molecular orbitals for the H2 molecule generated as a linear combination of

two s-functions, one on each nuclear centre.(6) Point group symmetry, notation and representations, and the group theoretical

condition for when an integral is zero.

I have elected to include a discussion of the variational principle and perturbationalmethods, although these are often covered in courses in elementary quantum mechan-ics. The properties of angular momentum coupling are used at the level of knowingthe difference between a singlet and triplet state. I do not believe that it is necessaryto understand the details of vector coupling to understand the implications.

Although I have tried to keep each chapter as self-contained as possible, there areunavoidable dependencies. The part in Chapter 3 describing HF methods is a prereq-uisite for understanding Chapter 4. Both these Chapters use terms and concepts forbasis sets which are treated in Chapter 5. Chapter 5, in turn, relies on concepts in Chap-ters 3 and 4, i.e. these three chapters form the core for understanding modern elec-tronic structure calculations. Many of the concepts in Chapters 3 and 4 are also usedin Chapters 6, 7, 9, 11 and 15 without further introduction, although these five chap-ters probably can be read with some benefits without a detailed understanding ofChapters 3 and 4. Chapter 8, and to a certain extent also Chapter 10, are fairly advancedfor an introductory textbook, such as the present, and can be skipped. They do,however, represent areas that are probably going to be more and more important inthe coming years. Function optimization, which is described separately in Chapter 14,is part of many areas, but a detailed understanding is not required for following thearguments in the other chapters. Chapters 12 and 13 are fairly self-contained, and formsome of the background for the methods in the other chapters. In my own course Inormally take Chapters 12, 13 and 14 fairly early in the course, as they provide back-ground for Chapters 3, 4 and 5.

If you would like to make comments, advise me of possible errors, make clarifica-tions, add references, etc., or view the current list of misprints and corrections, pleasevisit the author’s website (URL: http://bogense.chem.ou.dk/~icc).

xviii PREFACE TO THE FIRST EDITION

Preface to the Second Edition

The changes relative to the first edition are as follows:

• Numerous misprints and inaccuracies in the first edition have been corrected. Mostlikely some new ones have been introduced in the process, please check the bookwebsite for the most recent correction list and feel free to report possible problems.Since web addresses have a tendency to change regularly, please use your favouritesearch engine to locate the current URL.

• The methodologies and references in each chapter have been updated with newdevelopments published between 1998 and 2005.

• More extensive referencing. Complete referencing is impossible, given the largebreadth of subjects. I have tried to include references that preferably are recent,have a broad scope and include key references. From these the reader can get anentry into the field.

• Many figures and illustrations have been redone. The use of colour illustrations hasbeen deferred in favour of keeping the price of the book down.

• Each chapter or section now starts with a short overview of the methods, describedwithout mathematics. This may be useful for getting a feel for the methods, withoutembarking on all the mathematical details. The overview is followed by a moredetailed mathematical description of the method, including some key referenceswhich may be consulted for more details. At the end of the chapter or section, someof the pitfalls and the directions of current research are outlined.

• Energy units have been converted from kcal/mol to kJ/mol, based on the generalopinion that the scientific world should move towards SI units.

• Furthermore, some chapters have undergone major restructuring:

° Chapter 16 (Chapter 13 in the first edition) has been greatly expanded to includea summary of the most important mathematical techniques used in the book. Thegoal is to make the book more self-contained, i.e. relevant mathematical tech-niques used in the book are at least rudimentarily discussed in Chapter 16.

° All the statistical mechanics formalism has been collected in Chapter 13.° Chapter 14 has been expanded to cover more of the methodologies used in mole-

cular dynamics.

° Chapter 12 on optimization techniques has been restructured.° Chapter 6 on density functional methods has been rewritten.° A new Chapter 1 has been introduced to illustrate the similarities and differences

between classical and quantum mechanics, and to provide some fundamentalbackground.

° A rudimentary treatment of periodic systems has been incorporated in Chapters3 and 14.

° A new Chapter 17 has been introduced to describe statistics and QSAR methods.° I have tried to make the book more modular, i.e. each chapter is more self-con-

tained. This makes it possible to use only selected chapters, e.g. for a course, buthas the drawback of repeating the same things in several chapters, rather thansimply cross-referencing.

Although the modularity has been improved, there are unavoidable interdependen-cies. Chapters 3, 4 and 5 contain the essentials of electronic structure theory, and mostwould include Chapter 6 describing density functional methods. Chapter 2 contains adescription of empirical force field methods, and this is tightly coupled to the simula-tion methods in Chapter 14, which of course leans on the statistical mechanics inChapter 13. Chapter 1 on fundamental issues is of a more philosophical nature, andcan be skipped. Chapter 16 on mathematical techniques is mainly for those not alreadyfamiliar with this, and Chapter 17 on statistical methods may be skipped as well.

Definitions, acronyms and common phrases are marked in italic. In a change fromthe first edition, where underlining was used, italic text has also been used for empha-sizing important points.

A number of people have offered valuable help and criticisms during the updatingprocess. I would especially like to thank S. P. A. Sauer, H. J. Aa. Jensen, E. J. Baerendsand P. L. A. Popelier for having read various parts of the book and provided input.Remaining errors are of course my sole responsibility.

Specific comments on the preface to the first edition

Bohacek et al.1 have estimated the number of possible compounds composed of H, C,N, O and S atoms with 30 non-hydrogen atoms or fewer to be 1060. Although thisnumber is so large that only a very tiny fraction will ever be amenable to investiga-tion, the concept of functional groups means that one does not need to evaluate allcompounds in a given class to determine their properties. The number of alkanesmeeting the above criteria is ∼1010: clearly these will all have very similar and well-understood properties, and there is no need to investigate all 1010 compounds.

Reference

1. R. S. Bohacek, C. McMartin, W. C. Guida, Med. Res. Rev., 16 (1996), 3.

xx PREFACE TO THE SECOND EDITION

1 IntroductionChemistry is the science dealing with construction, transformation and properties ofmolecules.Theoretical chemistry is the subfield where mathematical methods are com-bined with fundamental laws of physics to study processes of chemical relevance.1

Molecules are traditionally considered as “composed” of atoms or, in a more generalsense, as a collection of charged particles, positive nuclei and negative electrons. Theonly important physical force for chemical phenomena is the Coulomb interactionbetween these charged particles. Molecules differ because they contain different nucleiand numbers of electrons, or because the nuclear centres are at different geometricalpositions. The latter may be “chemically different” molecules such as ethanol anddimethyl ether, or different “conformations” of for example butane.

Given a set of nuclei and electrons, theoretical chemistry can attempt to calculatethings such as:

• Which geometrical arrangements of the nuclei correspond to stable molecules?• What are their relative energies?• What are their properties (dipole moment, polarizability, NMR coupling constants,

etc.)?• What is the rate at which one stable molecule can transform into another?• What is the time dependence of molecular structures and properties?• How do different molecules interact?

The only systems that can be solved exactly are those composed of only one or twoparticles, where the latter can be separated into two pseudo one-particle problems byintroducing a “centre of mass” coordinate system. Numerical solutions to a given accu-racy (which may be so high that the solutions are essentially “exact”) can be gener-ated for many-body systems, by performing a very large number of mathematicaloperations. Prior to the advent of electronic computers (i.e. before 1950), the numberof systems that could be treated with a high accuracy was thus very limited. Duringthe sixties and seventies, electronic computers evolved from a few very expensive, dif-ficult to use, machines to become generally available for researchers all over the world.The performance for a given price has been steadily increasing since and the use ofcomputers is now widespread in many branches of science. This has spawned a new

Introduction to Computational Chemistry, Second Edition. Frank Jensen.© 2007 John Wiley & Sons, Ltd

field in chemistry, computational chemistry, where the computer is used as an “exper-imental” tool, much like, for example, an NMR spectrometer.

Computational chemistry is focused on obtaining results relevant to chemical prob-lems, not directly at developing new theoretical methods. There is of course a stronginterplay between traditional theoretical chemistry and computational chemistry.Developing new theoretical models may enable new problems to be studied, andresults from calculations may reveal limitations and suggest improvements in theunderlying theory. Depending on the accuracy wanted, and the nature of the systemat hand, one can today obtain useful information for systems containing up to severalthousand particles. One of the main problems in computational chemistry is selectinga suitable level of theory for a given problem, and to be able to evaluate the qualityof the obtained results. The present book will try to put the variety of modern com-putational methods into perspective, hopefully giving the reader a chance of estimat-ing which types of problems can benefit from calculations.

1.1 Fundamental IssuesBefore embarking on a detailed description of the theoretical methods in computa-tional chemistry, it may be useful to take a wider look at the background for the the-oretical models, and how they relate to methods in other parts of science, such asphysics and astronomy.

A very large fraction of the computational resources in chemistry and physics is usedin solving the so-called many-body problem. The essence of the problem is that two-particle systems can in many cases be solved exactly by mathematical methods,producing solutions in terms of analytical functions. Systems composed of more thantwo particles cannot be solved by analytical methods. Computational methods can,however, produce approximate solutions, which in principle may be refined to anydesired degree of accuracy.

Computers are not smart – at the core level they are in fact very primitive. Smartprogrammers, however, can make sophisticated computer programs, which may makethe computer appear smart, or even intelligent. But the basics of any computerprogram consist of a doing a few simple tasks such as:

• Performing a mathematical operation (adding, multiplying, square root, cosine, . . .)on one or two numbers.

• Determining the relationship (equal to, greater than, less than or equal to, . . .)between two numbers.

• Branching depending on a decision (add two numbers if N > 10, else subtract onenumber from the other).

• Looping (performing the same operation a number of times, perhaps on a set ofdata).

• Reading and writing data from and to external files.

These tasks are the essence of any programming language, although the syntax, datahandling and efficiency depend on the language. The main reason why computers areso useful is the sheer speed with which they can perform these operations. Even acheap off-the-shelf personal computer can perform billions (109) of operations persecond.

2 INTRODUCTION

Within the scientific world, computers are used for two main tasks: performingnumerically intensive calculations and analyzing large amounts of data. Such data can,for example, be pictures generated by astronomical telescopes or gene sequences inthe bioinformatics area that need to be compared. The numerically intensive tasks aretypically related to simulating the behaviour of the real world, by a more or less sophis-ticated computational model. The main problem in such simulations is the multi-scalenature of real-world problems, spanning from sub-nano to millimetres (10−10 − 10−3) inspatial dimensions, and from femto- to milliseconds (10−15 − 10−3) in the time domain.

1.2 Describing the SystemIn order to describe a system we need four fundamental features:

• System description – What are the fundamental units or “particles”, and how manyare there?

• Starting condition – Where are the particles and what are their velocities?• Interaction – What is the mathematical form for the forces acting between the

particles?• Dynamical equation – What is the mathematical form for evolving the system in

time?

The choice of “particles” puts limitations on what we are ultimately able to describe. Ifwe choose atomic nuclei and electrons as our building blocks, we can describe atoms and molecules, but not the internal structure of the atomic nucleus. If we chooseatoms as the building blocks, we can describe molecular structures, but not the details ofthe electron distribution.If we choose amino acids as the building blocks,we may be ableto describe the overall structure of a protein, but not the details of atomic movements.

1.2 DESCRIBING THE SYSTEM 3

Quarks

Electrons

Atoms Molecules Macro molecules

NucleiProtons Neutrons

Figure 1.1 Hierarchy of building blocks for describing a chemical system

The choice of starting conditions effectively determines what we are trying todescribe. The complete phase space (i.e. all possible values of positions and velocitiesfor all particles) is huge, and we will only be able to describe a small part of it. Ourchoice of starting conditions determines which part of the phase space we sample, forexample which (structural or conformational) isomer or chemical reaction we candescribe. There are many structural isomers with the molecular formula C6H6, but ifwe want to study benzene, we should place the nuclei in a hexagonal pattern, and startthem with relatively low velocities.

The interaction between particles in combination with the dynamical equation deter-mines how the system evolves in time. At the fundamental level, the only importantforce at the atomic level is the electromagnetic interaction. Depending on the choiceof system description (particles), however, this may result in different effective forces.

In force field methods, for example, the interactions are parameterized as stretch, bend,torsional, van der Waals, etc., interactions.

The dynamical equation describes the time evolution of the system. It is given as adifferential equation involving both time and space derivatives, with the exact formdepending on the particle masses and velocities. By solving the dynamical equation theparticles’ position and velocity can be predicted at later (or earlier) times relative tothe starting conditions, i.e. how the system evolves in the phase space.

1.3 Fundamental ForcesThe interaction between particles can be described in terms of either a force (F) or apotential (V). These are equivalent, as the force is the derivative of the potential withrespect to the position r.

(1.1)

Current knowledge indicates that there are four fundamental interactions, at leastunder normal conditions, as listed in Table 1.1.

F rVr

( ) = − ∂∂

4 INTRODUCTION

Table 1.1 Fundamental interactions

Name Particles Range (m) Relative strength

Strong interaction Quarks

between a proton and an electron, for example, the ratio between Velec and Vgrav is 1039.On a large macroscopic scale, such as planets, the situation is reversed. Here the grav-itational interaction completely dominates, and the electromagnetic interaction isabsent.

On a more fundamental level, it is believed that the four forces are really just dif-ferent manifestations of a single common interaction, because of the relatively lowenergy regime we are living in. It has been shown that the weak and electromagneticforces can be combined into a single unified theory, called quantum electrodynamics(QED). Similarly, the strong interaction can be coupled with QED into what is knownas the standard model. Much effort is being devoted to also include the gravitationalinteraction into a grand unified theory, and string theory is currently believed to holdthe greatest promise for such a unification.

Only the electromagnetic interaction is important at the atomic and molecular level, and in the large majority of cases, the simple Coulomb form (in atomic units) issufficient:

(1.3)

Within QED, the Coulomb interaction is only the zeroth-order term, and the completeinteraction can be written as an expansion in terms of the (inverse) velocity of light,c. For systems where relativistic effects are important (i.e. containing elements fromthe lower part of the periodic table), or when high accuracy is required, the first-ordercorrection (corresponding to an expansion up to 1/c2) for the electron–electron inter-action may be included:

(1.4)

The first-order correction is known as the Breit term, and a1 and a2 represent velocity operators. Physically, the first term in the Breit correction corresponds to mag-netic interaction between the two electrons, while the second term describes a “retar-dation” effect, since the interaction between distant particles is “delayed” relative tointeractions between close particles, owing to the finite value of c (in atomic units,c ~137).

1.4 The Dynamical EquationThe mathematical form for the dynamical equation depends on the mass and velocityof the particles, and can be divided into four regimes.

Newtonian mechanics, exemplified by Newton’s second law (F = ma), applies for“heavy”, “slow-moving” particles. Relativistic effects become important when thevelocity is comparable to the speed of light, causing an increase in the particle mass mrelative to the rest mass m0. A pragmatic borderline between Newtonian and rela-tivistic (Einstein) mechanics is ~1/3c, corresponding to a relativistic correction of a fewpercent.

Light particles display both wave and particle characteristics, and must be describedby quantum mechanics, with the borderline being approximately the mass of a proton.

V rr r

elec 1212

1 21 12 2 12

122

11

12

( ) = − ⋅ + ⋅( ) ⋅( )

r r

a a a a

V rCoulomb iji j

ij

q qr

( ) =

1.4 THE DYNAMICAL EQUATION 5

Electrons are much lighter and can only be described by quantum mechanics, whileatoms and molecules, with a few exceptions, behave essentially as classical particles.Hydrogen (protons), being the lightest nucleus, represents a borderline case, whichmeans that quantum corrections in some cases are essential. A prime example is thetunnelling of hydrogen through barriers, allowing reactions involving hydrogen tooccur faster than expected from transition state theory.

A major difference between quantum and classical mechanics is that classicalmechanics is deterministic while quantum mechanics is probabilistic (more correctly,quantum mechanics is also deterministic, but the interpretation is probabilistic).Deterministic means that Newton’s equation can be integrated over time (forward or backward) and can predict where the particles are at a certain time. This, forexample, allows prediction of where and when solar eclipses will occur many thou-sands of years in advance, with an accuracy of meters and seconds. Quantum mechan-ics, on the other hand, only allows calculation of the probability of a particle being ata certain place at a certain time. The probability function is given as the square of awave function, P(r,t) = Ψ2(r,t), where the wave function Ψ is obtained by solving eitherthe Schrödinger (non-relativistic) or Dirac (relativistic) equation. Although theyappear to be the same in Figure 1.2, they differ considerably in the form of the operator H.

For classical mechanics at low velocities compared with the speed of light, Newton’ssecond law applies.

(1.5)

If the particle mass is constant, the derivative of the momentum p is the mass timesthe acceleration.

(1.6)p v

Fp v

a

=

= = =

m

tm

tm

dd

dd

Fp= d

dt

6 INTRODUCTION

Velocity

Mass

Relativistic

Non-relativistic

Quantum Classical

~ 1/3 c~ 108 m/s

~ 10-27 kg ~ 1 amu

DiracHΨ= idΨ/dt

Schrödinger HΨ= idΨ/dt

Einstein F = ma

NewtonF = ma

Figure 1.2 Domains of dynamical equations

Since the force is the derivative of the potential (eq. (1.1)), and the acceleration is thesecond derivative of the position r with respect to time, it may also be written in a dif-ferential form.

(1.7)

Solving this equation gives the position of each particle as a function of time, i.e. r(t).At velocities comparable with the speed of light, Newton’s equation is formally

unchanged, but the particle mass becomes a function of the velocity, and the force istherefore not simply a constant (mass) times the acceleration.

(1.8)

For particles with small masses, primarily electrons, quantum mechanics must beemployed. At low velocities, the relevant equation is the time-dependent Schrödingerequation.

(1.9)

The Hamiltonian operator is given as a sum of kinetic and potential energy operators.

(1.10)

Solving the Schrödinger equation gives the wave function as a function of time, andthe probability of observing a particle at a position r and time t is given as the squareof the wave function.

(1.11)

For light particles moving at a significant fraction of the speed of light, the Schrödingerequation is replaced by the Dirac equation.

(1.12)

Although it is formally identical to the Schrödinger equation, the Hamiltonian opera-tor is significantly more complicated.

(1.13)

The a and b are 4 × 4 matrices, and the relativistic wave function consequently hasfour components.Traditionally, these are labelled the large and small components, eachhaving an a and b spin function (note the difference between the a and b matrices anda and b spin functions). The large component describes the electronic part of the wavefunction, while the small component describes the positronic (electron antiparticle)part of the wave function, and the a and b matrices couple these components. In thelimit of c → ∞, the Dirac equation reduces to the Schrödinger equation, and the two

H p VDirac = ⋅ +( ) +c mca b 2

HΨ Ψ= it

∂∂

P t tr r, ,( ) = ( )Ψ2

H T V

Tp

Schrodinger˙̇ = +

= = − ∇2

2

21

2m m

HΨ Ψ= it

∂∂

mm

v c=

−0

2 21

− =∂∂

∂∂

Vr

rm

t

2

2

1.4 THE DYNAMICAL EQUATION 7

large components of the wave function reduce to the a and b spin-orbitals in theSchrödinger picture.

1.5 Solving the Dynamical EquationBoth the Newton/Einstein and Schrödinger/Dirac dynamical equations are differen-tial equations involving the derivative of either the position vector or wave functionwith respect to time. For two-particle systems with simple interaction potentials V, these can be solved analytically, giving r(t) or Ψ(r,t) in terms of mathematical functions. For systems with more than two particles, the differential equation must be solved by numerical techniques involving a sequence of small finite time steps.

Consider a set of particles described by a position vector ri at a given time ti. A smalltime step ∆t later, the positions can be calculated from the velocities, acceleration,hyperaccelerations, etc., corresponding to a Taylor expansion with time as the variable.

(1.14)

The positions a small time step ∆t earlier were (replacing ∆t with −∆t)

(1.15)

Addition of these two equations gives a recipe for predicting the positions a time step∆t later from the current and previous positions, and the current acceleration, a methodknown as the Verlet algorithm.

(1.16)

Note that all odd terms in the Verlet algorithm disappear, i.e. the algorithm is correctto third order in the time step. The acceleration can be calculated from the force, orequivalently, the potential.

(1.17)

The time step ∆t is an important control parameter for a simulation. The largest valueof ∆t is determined by the fastest process occurring in the system, typically being anorder of magnitude smaller than the fastest process. For simulating nuclear motions,the fastest process is the motion of hydrogens, being the lightest particles. Hydrogenvibrations occur with a typical frequency of 3000cm−1, corresponding to ~1014 s−1, andtherefore necessitating time steps of the order of one femtosecond (10−15 s).

1.6 Separation of VariablesAs discussed in the previous section, the problem is solving a differential equation withrespect to either the position (classical) or wave function (quantum) for the particlesin the system. The standard method of solving differential equations is to find a set ofcoordinates where the differential equation can be separated into less complicatedequations. The first step is to introduce a centre of mass coordinate system, defined asthe mass-weighted sum of the coordinates of all particles, which allows the translation

aF V

r= = −

m m1 ∂

∂

r r r ai i i i t+ −= −( ) + ( ) +1 12

2 ∆ . . .

r r v a bi i i i it t t− = − ( ) + ( ) − ( ) +1 122 1

6

3∆ ∆ ∆ . . .

r r v a bi i i i it t t+ = + ( ) + ( ) + ( ) +1 122 1

6

3∆ ∆ ∆ . . .

8 INTRODUCTION

of the combined system with respect to a fixed coordinate system to be separated fromthe internal motion. For a two-particle system, the internal motion is described in termsof a reduced mass moving relative to the centre of mass, and this can be further trans-formed by introducing a coordinate system that reflects the symmetry of the interac-tion between the two particles. If the interaction only depends on the interparticledistance (e.g. Coulomb or gravitational interaction), the coordinate system of choiceis normally a polar (two-dimensional) or spherical polar (three-dimensional) system.In these coordinate systems, the dynamical equation can be transformed into solvingone-dimensional differential equations.

For more than two particles, it is still possible to make the transformation to thecentre of mass system. However, it is no longer possible to find a set of coordinatesthat allows a separation of the degrees of freedom for the internal motion, thus pre-venting an analytical solution. For many-body (N > 2) systems, the dynamical equationmust therefore be solved by computational (numerical) methods. Nevertheless, it isoften possible to achieve an approximate separation of variables based on physicalproperties, for example particles differing considerably in mass (such as nuclei andelectrons). A two-particle system consisting of one nucleus and one electron can besolved exactly by introducing a centre of mass system, thereby transforming theproblem into a pseudo-particle with a reduced mass (m = m1m2/(m1 + m2)) moving rel-ative to the centre of mass. In the limit of the nucleus being infinitely heavier than theelectron, the centre of mass system becomes identical to that of the nucleus. In thislimit, the reduced mass becomes equal to that of the electron which moves relative tothe (stationary) nucleus. For large, but finite, mass ratios, the approximation m ≈ me isunnecessary but may be convenient for interpretative purposes. For many-particlesystems, an exact separation is not possible, and the Born–Oppenheimer approxima-tion corresponds to assuming that the nuclei are infinitely heavier than the electrons.This allows the electronic problem to be solved for a given set of stationary nuclei.Assuming that the electronic problem can be solved for a large set of nuclear coordi-nates, the electronic energy forms a parametric hypersurface as a function of thenuclear coordinates, and the motion of the nuclei on this surface can then be solvedsubsequently.

If an approximate separation is not possible, the many-body problem can often betransformed into a pseudo one-particle system by taking the average interaction intoaccount. For quantum mechanics, this corresponds to the Hartree–Fock approxima-tion, where the average electron–electron repulsion is incorporated. Such pseudo one-particle solutions often form the conceptual understanding of the system, and providethe basis for more refined computational methods.

Molecules are sufficiently heavy that their motions can be described quite accuratelyby classical mechanics. In condensed phases (solution or solid state), there is a stronginteraction between molecules, and a reasonable description can only be attained byhaving a large number of individual molecules moving under the influence of eachother’s repulsive and attractive forces. The forces in this case are complex and cannotbe written in a simple form such as the Coulomb or gravitational interaction. No ana-lytical solutions can be found in this case, even for a two-particle (molecular) system.Similarly, no approximate solution corresponding to a Hartree–Fock model can be constructed. The only method in this case is direct simulation of the full dynamicalequation.

1.6 SEPARATION OF VARIABLES 9

1.6.1 Separating space and time variables

The time-dependent Schrödinger equation involves differentiation with respect toboth time and position, the latter contained in the kinetic energy of the Hamiltonianoperator.

(1.18)

For (bound) systems where the potential energy operator is time-independent (V(r,t) = V(r)), the Hamiltonian operator becomes time-independent and yields thetotal energy when acting on the wave function. The energy is a constant, independentof time, but depends on the space variables.

(1.19)

Inserting this in the time-dependent Schrödinger equation shows that the time andspace variables of the wave function can be separated.

(1.20)

The latter follows from solving the first-order differential equation with respect to time,and shows that the time dependence can be written as a simple phase factor multipliedwith the spatial wave function. For time-independent problems, this phase factor is nor-mally neglected, and the starting point is taken as the time-independent Schrödingerequation.

(1.21)

1.6.2 Separating nuclear and electronic variables

Electrons are very light particles and cannot be described by classical mechanics, whilenuclei are sufficiently heavy that they display only small quantum effects. The largemass difference indicates that the nuclear velocities are much smaller than the elec-tron velocities, and the electrons therefore adjust very fast to a change in the nucleargeometry.

For a general N-particle system, the Hamiltonian operator contains kinetic (T) andpotential (V) energy for all particles.

(1.22)

H T V

T T

V

= +

= = − ∇

∇ = ∂∂

+ ∂∂

+ ∂∂

=

= =

>

∑ ∑

∑

ii

N

ii

i

N

ii i i

iji j

N

m

x y z

V

1

2

1

22

2

2

2

2

2

12

H r r r r( ) ( ) = ( ) ( )Ψ ΨE

H r r r rr

r r

( ) ( ) = ( ) ( ) = ( )

( ) = ( ) −

Ψ Ψ Ψ

Ψ Ψ

, ,,

, e

t E t it

t

t iEt

∂∂

H r H r T r V r

H r r r r

,

, ,

t

t E t

( ) = ( ) = ( ) + ( )( ) ( ) = ( ) ( )Ψ Ψ

H r rr

H r T r V r

, ,,

, ,

t t it

t

t t

( ) ( ) = ( )

( ) = ( ) + ( )

Ψ Ψ∂∂

10 INTRODUCTION

The potential energy operator is the Coulomb potential (eq. (1.3)). Denoting nuclearcoordinates with R and subscript n, and electron coordinates with r and subscript e,this can be expressed as follows.

(1.23)

The above approximation corresponds to neglecting the coupling between the nuclearand electronic velocities, i.e. the nuclei are stationary from the electronic point of view.The electronic wave function thus depends parametrically on the nuclear coordinates,since it only depends on the position of the nuclei, not on their momentum. To a goodapproximation, the electronic wave function thus provides a potential energy surfaceupon which the nuclei move, and this separation is known as the Born–Oppenheimerapproximation.

The Born–Oppenheimer approximation is usually very good. For the hydrogen mol-ecule (H2) the error is of the order of 10−4 au, and for systems with heavier nuclei theapproximation becomes better. As we shall see later, it is possible only in a few casesto solve the electronic part of the Schrödinger equation to an accuracy of 10−4 au, i.e.neglect of the nuclear-electron coupling is usually only a minor approximation com-pared with other errors.

1.6.3 Separating variables in general

Assume that a set of variables can be found where the Hamiltonian operator H fortwo particles/variables can be separated into two independent terms, with each onlydepending on one particle/variable:

(1.24)

Assume furthermore that the Schrödinger equation for one particle/variable can besolved (exactly or approximately):

(1.25)

The solution to the two-particle problem can then be composed of solutions of one-variable Schrödinger equations.

(1.26)

This can be generalized to the case of N particles/variables:

(1.27)

H h=

=

=

∑∏∑

ii

ii

ii

E

Ψ f

e

Ψ == +

f fe e

1 2

1 2E

h i i i if e f=

H h h= +1 2

H R r R r

H H T

H T V V V

R r R R r

H R r R R r

T R R R

tot tot tot tot

tot e n

e e ne ee nn

tot n e

e e e e

n e n tot n

, ,

, ,

, ,

Ψ Ψ

Ψ Ψ ΨΨ Ψ

Ψ Ψ

( ) = ( )= += + + +

( ) = ( ) ( )( ) = ( ) ( )

+ ( )( ) ( ) = ( )

E

E

E E

1.6 SEPARATION OF VARIABLES 11

The properties in eq. (1.27) may be verified by inserting the entities in the Schrödingerequation (1.21).

1.7 Classical Mechanics1.7.1 The Sun–Earth system

The motion of the Earth around the Sun is an example of a two-body system that canbe treated by classical mechanics. The interaction between the two “particles” is thegravitational force.

(1.28)

The dynamical equation is Newton’s second law, which in differential form can bewritten as in eq. (1.29).

(1.29)

The first step is to introduce a centre of mass system, and the internal motion becomesmotion of a “particle” with a reduced mass given by eq. (1.30).

(1.30)

Since the mass of the Sun is 3 × 105 times larger than that of the Earth, the reducedmass is essentially identical to the Earth’s mass (m = 0.999997mEarth). To a very goodapproximation, the system can therefore be described as the Earth moving around theSun, which remains stationary.

The motion of the Earth around the Sun occurs in a plane, and a suitable coordi-nate system is a polar coordinate system (two-dimensional) consisting of r and q.

m =+

=+( )

≅M mM m

mm M

mSun EarthSun Earth

Earth

Earth SunEarth

1

− =∂∂

∂∂

Vr

rm

t

2

2

V r121 2

12

( ) = −C m mr

grav

12 INTRODUCTION

θ

rx = rcosθ y = rsinθ

y

x

Figure 1.3 A polar coordinate system

The interaction depends only on the distance r, and the differential equation(Newton’s equation) can be solved analytically.The bound solutions are elliptical orbitswith the Sun (more precisely, the centre of mass) at one of the foci, but for most ofthe planets, the actual orbits are close to circular. Unbound solutions corresponding tohyperbolas also exist, and could for example describe the path of a (non-returning)comet.

Each bound orbit can be classified in terms of the dimensions (largest and smallestdistance to the Sun), with an associated total energy. In classical mechanics, there areno constraints on the energy, and all sizes of orbits are allowed. If the zero point for

the energy is taken as the two particles at rest infinitely far apart, positive values ofthe total energy correspond to unbound solutions (hyperbolas, with the kinetic energybeing larger than the potential energy) while negative values correspond to boundorbits (ellipsoids, with the kinetic energy being less than the potential energy). Boundsolutions are also called stationary orbits, as the particle position returns to the samevalue with well-defined time intervals.

1.7.2 The solar system

Once we introduce additional planets in the Sun–Earth system, an analytical solutionfor the motions of all the planets can no longer be obtained. Since the mass of the Sunis so much larger than the remaining planets (the Sun is 1000 times heavier thanJupiter, the largest planet), the interactions between the planets can to a good approx-imation be neglected. For the Earth, for example, the second most important force isfrom the Moon, with a contribution that is 180 times smaller than that from the Sun.The next largest contribution is from Jupiter, being approximately 30000 times smaller(on average) than the gravitational force from the Sun. In this central field model, theorbit of each planet is determined as if it were the only planet in the solar system, andthe resulting computational task is a two-particle problem, i.e. elliptical orbits with theSun at one of the foci. The complete solar system is the unification of nine such orbits,and the total energy is the sum of all nine individual energies.

A formal refinement can be done by taking the average interaction between theplanets into account, i.e. a Hartree–Fock type approximation. In this model, the orbitof one planet (e.g. the Earth) is determined by taking the average interaction with allthe other planets into account. The average effect corresponds to spreading the massof the other planets evenly along their orbits.

The Hartree–Fock model represents only a very minute improvement over the inde-pendent orbit model for the solar system, since the planetary orbits do not cross. Theeffect of a planet inside the Earth’s orbit corresponds to adding its mass to the Sun,while the effect of the spread-out mass of a planet outside the Earth’s orbit is zero.The Hartree–Fock model for the Earth thus consists of increasing the Sun’s effectivemass with that of Mercury and Venus, i.e. a change of only 0.0003%. For the solarsystem there is thus very little difference between totally neglecting the planetaryinteractions and taking the average effect into account.

The real system, of course, includes all interactions, where each pair interactiondepends on the actual distance between the planets. The resulting planetary motions

1.7 CLASSICAL MECHANICS 13

Figure 1.4 Bound and unbound solutions to the classical two-body problem

cannot be solved analytically, but can be simulated numerically. From a given startingcondition, the system is allowed to evolve for many small time steps, and all interac-tions are considered constant within each time step. By sufficiently small time steps,this yields a very accurate model of the real many-particle dynamics, and will displaysmall wiggles of the planetary motion around the elliptical orbits calculated by eitherof the two independent-particle models.

Since the perturbations due to the other planets are significantly smaller than theinteraction with the Sun, the “wiggles” are small compared with the overall orbitalmotion, and a description of the solar system as planets orbiting the Sun in ellipticalorbits is a very good approximation to the true dynamics of the system.

1.8 Quantum Mechanics1.8.1 A hydrogen-like atom

A quantum analogue of the Sun–Earth system is a nucleus and one electron, i.e. ahydrogen-like atom. The force holding the nucleus and electron together is theCoulomb interaction.

14 INTRODUCTION

Figure 1.5 A Hartree–Fock model for the solar system

Figure 1.6 Modelling the solar system with actual interactions

(1.31)

The interaction again only depends on the distance, but owing to the small mass of theelectron, Newton’s equation must be replaced with the Schrödinger equation. Forbound states, the time-dependence can be separated out, as shown in Section 1.6.1,giving the time-independent Schrödinger equation.

(1.32)

The Hamiltonian operator for a hydrogen-like atom (nuclear charge of Z) can inCartesian coordinates and atomic units be written as eq. (1.33), with M being thenuclear and m the electron mass (m = 1 in atomic units).

(1.33)

The Laplace operator is given by eq. (1.34).

(1.34)

The two kinetic energy operators are already separated, since each only depends onthree coordinates. The potential energy operator, however, involves all six coordinates.The centre of mass system is introduced by the following six coordinates.

(1.35)

Here the X, Y, Z coordinates define the centre of mass system, and the x, y, z coordi-nates specify the relative position of the two particles. In these coordinates the Hamil-tonian operator can be rewritten as eq. (1.36).

(1.36)

The first term only involves the X, Y and Z coordinates, and the ∇2XYZ operator is obvi-ously separable in terms of X, Y and Z. Solution of the XYZ part gives translation ofthe whole system in three dimensions relative to the laboratory-fixed coordinatesystem. The xyz coordinates describe the relative motion of the two particles in termsof a pseudo-particle with a reduced mass m relative to the centre of mass.

(1.37)m =+

=+( )

≅M mM m

mm

M

mnuc elecnuc elec

elec

elec

nuc

elec

1

H = − ∇ − ∇ −+ +

12

2 22 2 2

12

XYZ xyzZ

x y zm

XMx mx

M mx x x

YMy my

M my y y

ZMz mz

M mz z z

= +( )+( )

= −

= +( )+( )

= −

= +( )+( )

= −

1 21 2

1 21 2

1 21 2

;

;

;

∇ = + +ii i ix y z

22

2

2

2

2

2

∂∂

∂∂

∂∂

H = − ∇ − ∇ −−( ) + −( ) + −( )

12

12

12

22

1 22

1 22

1 22M m

Z

x x y y z z

HΨ Ψ= E

V r121 2

12

( ) = q qr

1.8 QUANTUM MECHANICS 15

The potential energy becomes very simple, but the kinetic energy operator becomescomplicated.

(1.38)

The kinetic energy operator,however, is almost separable in spherical polar coordinates,and the actual method of solving the differential equation can be found in a number oftextbooks. The bound solutions (negative total energy) are called orbitals and can beclassified in terms of three quantum numbers, n, l and m, corresponding to the threespatial variables r, q and j. The quantum numbers arise from the boundary conditionson the wave function, i.e. it must be periodic in the q and j variables, and must decay tozero as r → ∞. Since the Schrödinger equation is not completely separable in sphericalpolar coordinates, there exist the restrictions n > l ≥ |m|.The n quantum number describesthe size of the orbital, the l quantum number describes the shape of the orbital, while them quantum number describes the orientation of the orbital relative to a fixed coordinatesystem.The l quantum number translates into names for the orbitals:

• l = 0 : s-orbital• l = 1 : p-orbital• l = 2 : d-orbital, etc.

The orbitals can be written as a product of a radial function, describing the behaviourin terms of the distance r between the nucleus and electron, and spherical harmonicfunctions Ylm representing the angular part in terms of the angles q and j. The orbitalscan be visualized by plotting three-dimensional objects corresponding to the wavefunction having a specific value, e.g. Ψ2 = 0.10.

H = − ∇ −

∇ = + +

12

1 1 1

2

22

22 2 2

2

2

m∂∂

∂∂ q

∂∂q

q ∂∂q q

∂∂j

qj

qj

r

r

Zr

r rr

r r rsinsin

sin

16 INTRODUCTION

r

ϕ

θ

x

y

z

x = rsinθ cosϕy = rsinθ sinϕ z = rcosθ

Figure 1.7 A spherical polar coordinate system

For the hydrogen atom, the nucleus is approximately 1800 times heavier than the electron, giving a reduced mass of 0.9995melec. Similar to the Sun–Earth system, thehydrogen atom can therefore to a good approximation be considered as an electronmoving around a stationary nucleus, and for heavier elements the approximationbecomes better (with a uranium nucleus, for example, the nucleus/electron mass ratiois ~430000). Setting the reduced mass equal to the electron mass corresponds tomaking the assumption that the nucleus is infinitely heavy and therefore stationary.

The potential energy again only depends on the distance between the two particles,but in contrast to the Sun–Earth system, the motion occurs in three dimensions, andit is therefore advantageous to transform the Schrödinger equation into a sphericalpolar set of coordinates.

The orbitals for different quantum numbers are orthogonal and can be chosen to benormalized.

(1.39)

The orthogonality of the orbitals in the angular part (l and m quantum numbers)follows from the shape of the spherical harmonic functions, as these have l nodal planes(points where the wave function is zero). The orthogonality in the radial part (nquantum number) is due to the presence of (n–l–1) radial nodes in the wave function.

In contrast to classical mechanics, where all energies are allowed, wave functionsand associated energies are quantized, i.e. only certain values are allowed. The energyonly depends on n for a given nuclear charge Z, and is given by eq. (1.40).

(1.40)

Unbound solutions have a positive total energy and correspond to scattering of anelectron by the nucleus.

1.8.2 The helium atom

Like the solar system, it is not possible to find a set of coordinates where theSchrödinger equation can be solved analytically for more than two particles (i.e. formany-electron atoms). Owing to the dominance of the Sun’s gravitational field, acentral field approximation provides a good description of the actual solar system, butthis is not the case for an atomic system.The main differences between the solar systemand an atom such as helium are:

(1) The interaction between the electrons is only a factor of two smaller than betweenthe nucleus and electrons, compared with a ratio of at least 1000 for the solarsystem.

(2) The electron–electron interaction is repulsive, compared with the attractionbetween planets.

EZn

= −2

22

Ψ Ψn l m n l m n n l l m m, , , , , , ,′ ′ ′ ′ ′ ′= d d d

1.8 QUANTUM MECHANICS 17

Table 1.2 Hydrogenic orbitals obtained from solving the Schrödinger equation

n l m Ψn,l,m(r,q,j) Shape and size

1 0 0 Y0,0(q,j)e−Zr

2 0 0 Y0,0(q,j)(2 − Zr)e−Zr/2

1 ±1, 0 Y1,m(q,j)Zre−Zr/2

3 0 0 Y0,0(q,j)(27 − 18Zr + 2Z2r2)e−Zr/3

1 ±1, 0 Y1,m(q,j)Zr(6 − Zr)e−Zr/3

2 ±2, ±1, 0 Y2,m(q,j)Z2r2e−Zr/3

(3) The motion of the electrons must be described by quantum mechanics owing tothe small electron mass, and the particle position is determined by an orbital, thesquare of which gives the probability of finding the electron at a given position.

(4) Electrons are indistinguishable particles having a spin of 1/2. This fermion charac-ter requires the total wave function to be antisymmetric, i.e. it must change signwhen interchanging two electrons. The antisymmetry results in the so-calledexchange energy, which is a non-classical correction to the Coulomb interaction.

The simplest atomic model would be to neglect the electron–electron interaction, andonly take the nucleus–electron attraction into account. In this model each orbital forthe helium atom is determined by solving a hydrogen-like system with a nucleus andone electron, yielding hydrogen-like orbitals, 1s, 2s, 2p, 3s, 3p, 3d, etc., with Z = 2. Thetotal wave function is obtained from the resulting orbitals subject to the aufbau andPauli principles. These principles say that the lowest energy orbitals should be filledfirst and only two electrons (with different spin) can occupy each orbital, i.e. the elec-tron configuration becomes 1s2. The antisymmetry condition is conveniently fulfilledby writing the total wave function as a Slater determinant, since interchanging any tworows or columns changes the sign of the determinant. For a helium atom, this wouldgive the following (unnormalized) wave function, with the orbitals given in Table 1.2with Z = 2.

(1.41)

The total energy calculated by this wave function is simply twice the orbital energy,−4.000 au, which is in error by 38% compared with the experimental value of −2.904au. Alternatively, we can use the wave function given by eq. (1.41), but include the electron–electron interaction in the energy calculation, giving a value of −2.750 au.

A better approximation can be obtained by taking the average repulsion between theelectrons into account when determining the orbitals, a procedure known as theHartree–Fock approximation. If the orbital for one of the electrons were somehowknown, the orbital for the second electron could be calculated in the electric field of thenucleus and the first electron, described by its orbital. This argument could just as wellbe used for the second electron with respect to the first electron.The goal is therefore tocalculate a set of self-consistent orbitals, and this can be done by iterative methods.

For the solar system, the non-crossing of the planetary orbitals makes theHartree–Fock approximation only a very minor improvement over a central fieldmodel. For a many-electron atom, however, the situation is different since the positionof the electrons is described by three-dimensional probability functions (square of theorbitals), i.e. the electron “orbits” “cross”. The average nucleus–electron distance foran electron in a 2s-orbital is larger than for one in a 1s-orbital, but there is a finiteprobability that a 2s-electron is closer to the nucleus than a 1s-electron. If the 1s-electrons in lithium were completely inside the 2s-orbital, the latter would experiencean effective nuclear charge of 1.00, but owing to the 2s-electron penetrating the 1s-orbital, the effective nuclear charge for an electron in a 2s-orbital is 1.26. The 2s-elec-tron in return screens the nuclear charge felt by the 1s-electrons, making the effectivenuclear charge felt by the 1s-electrons less than 3.00. The mutual screening of the two 1s-electrons in helium produces an effective nuclear charge of 1.69, yielding a total

Φ =( )( )

( )( )

= ( ) ( ) − ( ) ( )ff

ff

f f f faa

b

ba b b a

1

1

1

11 1 1 1

1

2

1

21 2 1 2s

s

s

ss s s s

18 INTRODUCTION

The equal mass of all the electrons and the strong interaction between them makesthe Hartree–Fock model less accurate than desirable, but it is still a big improvementover an independent orbital model. The Hartree–Fock model typically accounts for~99% of the total energy, but the remaining correlation energy is usually very impor-tant for chemical purposes. The correlation between the electrons describes the“wiggles” relative to the Hartree–Fock orbitals due to the instantaneous interactionbetween the electrons, rather than just the average repulsion. The goal of correlatedmethods for solving the Schrödinger equation is to calculate the remaining correctiondue to the electron–electron interaction.

1.9 ChemistryThe Born–Oppenheimer separation of the electronic and nuclear motions is a cor-nerstone in computational chemistry. Once the electronic Schrödinger equation hasbeen solved for a large number of nuclear geometries (and possibly also for severalelectronic states), the potential energy surface (PES) is known.The motion of the nucleion the PES can then be solved either classically (Newton) or by quantum(Schrödinger) methods. If there are N nuclei, the dimensionality of the PES is 3N, i.e.there are 3N nuclear coordinates that define the geometry. Of these coordinates, threedescribe the overall translation of the molecule, and three describe the overall rota-tion of the molecule with respect to three axes. For a linear molecule, only two coor-dinates are necessary for describing the rotation. This leaves 3N − 6(5) coordinates todescribe the internal movement of the nuclei, which for small displacements may bechosen as “vibrational normal coordinates”.

1.9 CHEMISTRY 19

Table 1.3 Helium atomic energies in various approximations

Wave function Zeff Energy (au)

He+ exponential orbital, no electron–electron repulsion 2.00 −4.000He+ exponential orbital, including electron–electron repulsion 2.00 −2.750Optimum single exponential orbital 1.69 −2.848Best orbital, Hartree–Fock limit −2.862Experimental −2.904

energy of −2.848 au, which is a significant improvement relative to the model withorbitals employing a fixed nuclear charge of 2.00.

Although the effective nuclear charge of 1.69 represents the lowest possible energywith the functional form of the orbitals in Table 1.2, it is possible to further refine themodel by relaxing the functional form of the orbitals from a strict exponential.Although the exponential form is the exact solution for a hydrogen-like system, thisis not the case for a many-electron atom. Allowing the orbitals to adopt best possibleform, and simultaneously optimizing the exponents (“effective nuclear charge”), givesan energy of −2.862 au. This represents the best possible independent-particle modelfor the helium atom, and any further refinement must include the instantaneous cor-relation between the electrons. By using the electron correlation methods described inChapter 4, it is possible to reproduce the experimental energy of −2.904 au.

It should be stressed that nuclei are heavy enough that quantum effects are almostnegligible, i.e. they behave to a good approximation as classical particles. Indeed, ifnuclei showed significant quantum aspects, the concept of molecular structure (i.e. dif-ferent configurations and conformations) would not have any meaning, since the nucleiwould simply tunnel through barriers and end up in the global minimum. Dimethylether, for example, would spontaneously transform into ethanol. Furthermore, it wouldnot be possible to speak of a molecular geometry, since the Heisenberg uncertaintyprinciple would not permit a measure of nuclear positions with an accuracy smallerthan the molecular dimension.

Methods aimed at solving the electronic Schrödinger equation are broadly referredto as “electronic structure calculations”. An accurate determination of the electronicwave function is very demanding. Constructing a complete PES for molecules con-taining more than three or four atoms is virtually impossible. Consider, for example,mapping the PES by calculating the electronic energy for every 0.1Å over say a 1Årange (a very coarse mapping). With three atoms, there are three internal coordinates,giving 103 points to be calculated. Four atoms already produce six internal coordinates,giving 106 points, which is possible to calculate, but only with a very determined effort.Larger systems are out of reach. Constructing global PES’s for all but the smallest mol-ecules is thus impossible. By restricting the calculations to the “chemically interesting”part of the PES, however, it is possible to obtain useful information. The interestingparts of a PES are usually nuclear arrangements that have low energies. For example,nuclear movements near a minimum on the PES, which corresponds to a stable mol-ecule, are molecular vibrations. Chemical reactions correspond to larger movements,and may in the simplest approximation be described by locating the lowest energy pathleading from one minimum on the PES to another.

These considerations lead to the following definition:

Chemistry is knowing the energy as a function of the nuclear coordinates.

The large majority of what are commonly referred to as molecular properties may sim-ilarly be defined as:

Properties are knowing how the energy changes upon adding a perturbation.

In the following we will look at some aspects of solving the electronic Schrödingerequation or otherwise construct a PES, how to deal with the movement of nuclei onthe PES, and various technical points of commonly used methods. A word of cautionhere: although it is the nuclei that move, and the electrons follow “instantly” (accord-ing to the Born–Oppenheimer approximation), it is common also to speak of “atoms”moving. An isolated atom consists of a nucleus and some electrons, but in a moleculethe concept of an atom is not well defined. Analogously to the isolated atom, an atomin a molecule should consist of a nucleus and some electrons. But how does one par-tition the total electron distribution in a molecule such that a given portion belongs to a given nucleus? Nevertheless, the words nucleus and atom are often used interchangeably.

Much of the following will concentrate on describing individual molecules. Experi-ments are rarely done on a single molecule; rather they are performed on macroscopicsamples with perhaps 1020 molecules. The link between the properties of a single mol-ecule, or a small collection of molecules, and the macroscopic observable is statistical

20 INTRODUCTION