E-Book
78,99 €

Introduction to Computational Chemistry E-Book

Frank Jensen

4,5

78,99 €

Sammeln Sie Punkte in unserem Gutscheinprogramm und kaufen Sie E-Books und Hörbücher mit bis zu 100% Rabatt.

Mehr erfahren.

Herausgeber: John Wiley & Sons
Kategorie: Wissenschaft und neue Technologien
Sprache: Englisch

Beschreibung

Introduction to Computational Chemistry 3rd Edition provides a comprehensive account of the fundamental principles underlying different computational methods. Fully revised and updated throughout to reflect important method developments and improvements since publication of the previous edition, this timely update includes the following significant revisions and new topics:

Polarizable force fields
Tight-binding DFT
More extensive DFT functionals, excited states and time dependent molecular properties
Accelerated Molecular Dynamics methods
Tensor decomposition methods
Cluster analysis
Reduced scaling and reduced prefactor methods

Additional information is available at: www.wiley.com/go/jensen/computationalchemistry3

Details

Sie lesen das E-Book in den Legimi-Apps auf:

Android

iOS

von Legimi
zertifizierten E-Readern

Seitenzahl: 1499

Veröffentlichungsjahr: 2016

Bewertungen

4,5 (16 Bewertungen)

Rezensionen(0 Rezensionen)

Leseprobe

Introduction to Computational Chemistry

Third Edition

Frank Jensen

Department of Chemistry, Aarhus University, Denmark

Registered Office:

John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK

Editorial Offices:

9600 Garsington Road, Oxford, OX4 2DQ, UK The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK 111 River Street, Hoboken, NJ 07030-5774, USA

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/wiley-blackwell.

The right of the author to be identified as the author of this work has been asserted in accordance with the UK Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.

Limit of Liability/Disclaimer of Warranty: While the publisher and author(s) have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Names: Jensen, Frank, author. Title: Introduction to computational chemistry / Frank Jensen. Description: Third edition. | Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2017. | Includes index. Identifiers: LCCN 2016039772 (print) | LCCN 2016052630 (ebook) | ISBN 9781118825990 (pbk.) | ISBN 9781118825983 (pdf) | ISBN 9781118825952 (epub) Subjects: LCSH: Chemistry, Physical and theoretical–Data processing. | Chemistry, Physical and theoretical–Mathematics. Classification: LCC QD455.3.E4 J46 2017 (print) | LCC QD455.3.E4 (ebook) | DDC 541.0285–dc23 LC record available at https://lccn.loc.gov/2016039772

A catalogue record for this book is available from the British Library.

ISBN: 9781118825990

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Cover Page

Title Page

Preface to the First Edition

Preface to the Second Edition

Specific Comments on the Preface to the First Edition

Reference

Preface to the Third Edition

Reference

1 Introduction

1.1 Fundamental Issues

1.2 Describing the System

1.3 Fundamental Forces

1.4 The Dynamical Equation

1.5 Solving the Dynamical Equation

1.6 Separation of Variables

1.7 Classical Mechanics

1.8 Quantum Mechanics

1.9 Chemistry

References

2 Force Field Methods

2.1 Introduction

2.2 The Force Field Energy

2.3 Force Field Parameterization

2.4 Differences in Atomistic Force Fields

2.5 Water Models

2.6 Coarse Grained Force Fields

2.7 Computational Considerations

2.8 Validation of Force Fields

2.9 Practical Considerations

2.10 Advantages and Limitations of Force Field Methods

2.11 Transition Structure Modeling

2.12 Hybrid Force Field Electronic Structure Methods

References

3 Hartree–Fock Theory

3.1 The Adiabatic and Born–Oppenheimer Approximations

3.2 Hartree–Fock Theory

3.3 The Energy of a Slater Determinant

3.4 Koopmans' Theorem

3.5 The Basis Set Approximation

3.6 An Alternative Formulation of the Variational Problem

3.7 Restricted and Unrestricted Hartree–Fock

3.8 SCF Techniques

3.9 Periodic Systems

References

4 Electron Correlation Methods

4.1 Excited Slater Determinants

4.2 Configuration Interaction

4.3 Illustrating how CI Accounts for Electron Correlation, and the RHF Dissociation Problem

4.4 The UHF Dissociation and the Spin Contamination Problem

4.5 Size Consistency and Size Extensivity

4.6 Multiconfiguration Self-Consistent Field

4.7 Multireference Configuration Interaction

4.8 Many-Body Perturbation Theory

4.9 Coupled Cluster

4.10 Connections between Coupled Cluster, Configuration Interaction and Perturbation Theory

4.11 Methods Involving the Interelectronic Distance

4.12 Techniques for Improving the Computational Efficiency

4.13 Summary of Electron Correlation Methods

4.14 Excited States

4.15 Quantum Monte Carlo Methods

References

5 Basis Sets

5.1 Slater- and Gaussian-Type Orbitals

5.2 Classification of Basis Sets

5.3 Construction of Basis Sets

5.4 Examples of Standard Basis Sets

5.5 Plane Wave Basis Functions

5.6 Grid and Wavelet Basis Sets

5.7 Fitting Basis Sets

5.8 Computational Issues

5.9 Basis Set Extrapolation

5.10 Composite Extrapolation Procedures

5.11 Isogyric and Isodesmic Reactions

5.12 Effective Core Potentials

5.13 Basis Set Superposition and Incompleteness Errors

References

6 Density Functional Methods

6.1 Orbital-Free Density Functional Theory

6.2 Kohn–Sham Theory

6.3 Reduced Density Matrix and Density Cumulant Methods

6.4 Exchange and Correlation Holes

6.5 Exchange–Correlation Functionals

6.6 Performance of Density Functional Methods

6.7 Computational Considerations

6.8 Differences between Density Functional Theory and Hartree-Fock

6.9 Time-Dependent Density Functional Theory (TDDFT)

6.10 Ensemble Density Functional Theory

6.11 Density Functional Theory Problems

6.12 Final Considerations

References

7 Semi-empirical Methods

7.1 Neglect of Diatomic Differential Overlap (NDDO) Approximation

7.2 Intermediate Neglect of Differential Overlap (INDO) Approximation

7.3 Complete Neglect of Differential Overlap (CNDO) Approximation

7.4 Parameterization

7.5 Hückel Theory

7.6 Tight-Binding Density Functional Theory

7.7 Performance of Semi-empirical Methods

7.8 Advantages and Limitations of Semi-empirical Methods

References

8 Valence Bond Methods

8.1 Classical Valence Bond Theory

8.2 Spin-Coupled Valence Bond Theory

8.3 Generalized Valence Bond Theory

References

9 Relativistic Methods

9.1 The Dirac Equation

9.2 Connections between the Dirac and Schrödinger Equations

9.3 Many-Particle Systems

9.4 Four-Component Calculations

9.5 Two-Component Calculations

9.6 Relativistic Effects

References

10 Wave Function Analysis

10.1 Population Analysis Based on Basis Functions

10.2 Population Analysis Based on the Electrostatic Potential

10.3 Population Analysis Based on the Electron Density

10.4 Localized Orbitals

10.5 Natural Orbitals

10.6 Computational Considerations

10.7 Examples

References

11 Molecular Properties

11.1 Examples of Molecular Properties

11.2 Perturbation Methods

11.3 Derivative Techniques

11.4 Response and Propagator Methods

11.5 Lagrangian Techniques

11.6 Wave Function Response

11.7 Electric Field Perturbation

11.8 Magnetic Field Perturbation

11.9 Geometry Perturbations

11.10 Time-Dependent Perturbations

11.11 Rotational and Vibrational Corrections

11.12 Environmental Effects

11.13 Relativistic Corrections

References

12 Illustrating the Concepts

12.1 Geometry Convergence

12.2 Total Energy Convergence

12.3 Dipole Moment Convergence

12.4 Vibrational Frequency Convergence

12.5 Bond Dissociation Curves

12.6 Angle Bending Curves

12.7 Problematic Systems

12.8 Relative Energies of C4H6 Isomers

References

13 Optimization Techniques

13.1 Optimizing Quadratic Functions

13.2 Optimizing General Functions: Finding Minima

13.3 Choice of Coordinates

13.4 Optimizing General Functions: Finding Saddle Points (Transition Structures)

13.5 Constrained Optimizations

13.6 Global Minimizations and Sampling

13.7 Molecular Docking

13.8 Intrinsic Reaction Coordinate Methods

References

14 Statistical Mechanics and Transition State Theory

14.1 Transition State Theory

14.2 Rice–Ramsperger–Kassel–Marcus Theory

14.3 Dynamical Effects

14.4 Statistical Mechanics

14.5 The Ideal Gas, Rigid-Rotor Harmonic-Oscillator Approximation

14.6 Condensed Phases

References

15 Simulation Techniques

15.1 Monte Carlo Methods

15.2 Time-Dependent Methods

15.3 Periodic Boundary Conditions

15.4 Extracting Information from Simulations

15.5 Free Energy Methods

15.6 Solvation Models

References

16 Qualitative Theories

16.1 Frontier Molecular Orbital Theory

16.2 Concepts from Density Functional Theory

16.3 Qualitative Molecular Orbital Theory

16.4 Energy Decomposition Analyses

16.5 Orbital Correlation Diagrams: The Woodward–Hoffmann Rules

16.6 The Bell–Evans–Polanyi Principle/Hammond Postulate/Marcus Theory

16.7 More O'Ferrall–Jencks Diagrams

References

17 Mathematical Methods

17.1 Numbers, Vectors, Matrices and Tensors

17.2 Change of Coordinate System

17.3 Coordinates, Functions, Functionals, Operators and Superoperators

17.4 Normalization, Orthogonalization and Projection

17.5 Differential Equations

17.6 Approximating Functions

17.7 Fourier and Laplace Transformations

17.8 Surfaces

References

18 Statistics and QSAR

18.1 Introduction

18.2 Elementary Statistical Measures

18.3 Correlation between Two Sets of Data

18.4 Correlation between Many Sets of Data

18.5 Quantitative Structure–Activity Relationships (QSAR)

18.6 Non-linear Correlation Methods

18.7 Clustering Methods

References

19 Concluding Remarks

Appendix A

Notation

Appendix B

The Variational Principle

The Hohenberg–Kohn Theorems

The Adiabatic Connection Formula

Reference

Appendix C

Atomic Units

Appendix D

Z-Matrix Construction

Appendix E

First and Second Quantization

References

Index

WILEY END USER LICENSE AGREEMENT

Pages

xvi

xvii

xviii

xix

xxi

xxii

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

377

378

379

380

381

382

383

384

385

386

387

388

389

391

393

394

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

531

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

List of Tables

Appendix C

Table C.1

Chapter 1

Table 1.1

Table 1.2

Table 1.3

Chapter 2

Table 2.1

Table 2.2

Table 2.3

Table 2.4

Table 2.5

Table 2.6

Table 2.7

Table 2.8

Table 2.9

Chapter 4

Table 4.1

Table 4.2

Table 4.3

Table 4.4

Table 4.5

Table 4.6

Table 4.7

Chapter 5

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Chapter 6

Table 6.1

Table 6.2

Chapter 7

Table 7.1

Table 7.2

Table 7.3

Chapter 8

Table 8.1

Chapter 9

Table 9.1

Chapter 10

Table 10.1

Table 10.2

Table 10.3

Chapter 11

Table 11.1

Table 11.2

Table 11.3

Chapter 12

Table 12.1

Table 12.2

Table 12.3

Table 12.4

Table 12.5

Table 12.6

Table 12.7

Table 12.8

Table 12.9

Table 12.10

Table 12.11

Table 12.12

Table 12.13

Table 12.14

Table 12.15

Table 12.16

Table 12.17

Table 12.18

Table 12.19

Table 12.20

Chapter 13

Table 13.1

Table 13.2

Chapter 14

Table 14.1

Table 14.2

Table 14.3

Chapter 15

Table 15.1

Table 15.2

Chapter 16

Table 16.1

Table 16.2

Chapter 17

Table 17.1

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. For the newcomer to the field, there are three main problems:

Deciphering the code. The language of computational chemistry is littered with acronyms, what do these abbreviations stand for in terms of underlying assumptions and approximations?

Technical problems. How does one actually run the program and what to look for in the output?

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 live in another country, you have to learn the language. If you want to use computational chemistry methods, you need to learn the acronyms. I have tried in the present book to include a good fraction of the most commonly used abbreviations and standard procedures.

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

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

The ease with which calculations can be performed means that point (3) has become the central theme in computational chemistry. It is quite easy to run a series of calculations that produce results that are absolutely meaningless. The program will not tell you whether the chosen method is valid for the problem you are studying. Quality assessment is thus an absolute requirement. This, however, requires much more experience and insight than just running the program. A basic understanding of the theory behind the method is needed, and a knowledge of the performance of the method for other 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 computational chemistry has (had) a somewhat bleak reputation. “If five different computational methods give five widely different results, what has computational chemistry contributed? You just pick the number closest to experiments and claim that you can reproduce experimental data accurately.” One commonly sees statements of the type “The theoretical results for property X are in disagreement. Calculation at the CCSD(T)/6-31G(d,p) level predicts that…, while the MINDO/3 method gives opposing results. There is thus no clear consent from theory.” This is clearly a lack of understanding of the quality of the calculations. If the results disagree, there is a very high probability that the CCSD(T) results are basically correct, and the MINDO/3 results are 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 difficult task in computational chemistry. I hope the present book will give some idea of the 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 competitive with experimental methods for generating a value for a given property of a given molecule, the number of possible molecules (there are an estimated 10200 molecules with a molecular weight less than 850) and their associated properties is so huge that only a very tiny fraction will ever be amenable to calculations (or experiments). Furthermore, with the constant increase in computational power, a calculation that barely 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 (with respect to theoretical level) will typically only have a lifetime of a few years before being surpassed by more accurate calculations.

The real strength of computational chemistry is the ability to generate data (e.g. by analyzing the wave function) from which a human may gain insight, and thereby rationalize the behavior of a large class of molecules. Such insights and rationalizations are much more likely to be useful over a longer period of time than the raw results themselves. A good example is the concept used by organic chemists with molecules composed of functional groups, and representing reactions by “pushing electrons”. This may not be particularly accurate from a quantum mechanical point of view, but it is very effective in rationalizing a large body of experimental results, and has good predictive power.

Just as computers do not solve problems, mathematics by itself does not provide insight. It merely provides formulas, a framework for organizing thoughts. It is in this spirit that I have tried to write this book. Only the necessary (obviously a subjective criterion) mathematical background has been provided, the aim being that the reader should be able to understand the premises and limitations of different methods, and follow the main steps in running a calculation. This means that in many cases I have omitted 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 stringent and detailed derivation of the mathematics. The goal of this book is to provide an overview of commonly used methods, giving enough theoretical background to understand why, for example, the AMBER force field is used for modeling proteins but MM2 is used for small organic molecules, or why coupled cluster inherently is an iterative method, while perturbation theory and configuration interaction inherently are non-iterative methods, although the CI problem in practice is solved by iterative techniques.

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, predicting 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 most methods that are already extensively used, together with some that I expect to become generally available in the near future. How detailed the methods are described depends partly on how practical and commonly used the methods are (both in terms of computational resources and software), and partly reflects my own limitations in terms of understanding. Although simulations (e.g. molecular dynamics) are becoming increasingly 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 methods must be translated into working computer programs in order to produce results. Different algorithms, however, may have different behaviors in practice, and it becomes necessary to be able to evaluate whether a certain type of calculation can be carried out with the available computers. The book thus contains some guidelines for evaluating what type of resources are necessary for carrying out a given calculation.

The present book grew out of a series of lecture notes that I have used for teaching a course in computational chemistry at Odense University, and the style of the book reflects its origin. It is difficult to master all disciplines in the vast field of computational 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. Remaining errors are of course my sole responsibility. A good part of the final transformation from a set of lecture notes to the present book was done during a sabbatical leave spent with Prof. L. Radom at the Research School of Chemistry, Australia National University, 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 common phrases are marked in italic; these can be found in the index. Underline is used for emphasizing 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 possible, different branches in computational chemistry often use different symbols for the same quantity. In order to comply with common usage, I have elected sometimes to 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 denoted F when discussing vibrations, and called the Hessian H for optimization purposes.

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

The Schrödinger equation, with the consequences of quantized solutions and quantum numbers.

The interpretation of the square of the wave function as a probability distribution, the Heisenberg uncertainty principle and the possibility of tunneling.

The solutions for the hydrogen atom, atomic orbitals.

The solutions for the harmonic oscillator and rigid rotor.

The molecular orbitals for the H

molecule generated as a linear combination of two

-functions, one on each nuclear centre.

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 perturbational methods, although these are often covered in courses in elementary quantum mechanics. The properties of angular momentum coupling are used at the level of knowing the difference between a singlet and triplet state. I do not believe that it is necessary to understand the details of vector coupling to understand the implications.

Although I have tried to keep each chapter as self-contained as possible, there are unavoidable dependencies. The part in Chapter 3 describing HF methods is a prerequisite for understanding Chapter 4. Both these chapters use terms and concepts for basis sets which are treated in Chapter 5. Chapter 5, in turn, relies on concepts in Chapters 3 and 4, that is these three chapters form the core for understanding modern electronic structure calculations. Many of the concepts in Chapters 3 and 4 are also used in Chapters 6, 7, 9, 11 and 15 without further introduction, although these five chapters probably can be read with some benefits without a detailed understanding of Chapters 3 and 4. Chapter 8, and to a certain extent also Chapter 10, are fairly advanced for 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 in the 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 the arguments in the other chapters. Chapters 12 and 13 are fairly self-contained, and form some of the background for the methods in the other chapters. In my own course I normally take Chapters 12, 13 and 14 fairly early in the course, as they provide background for Chapters 3, 4 and 5.

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

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. Most likely some new ones have been introduced in the process; please check the book website 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 favourite search engine to locate the current URL.

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

More extensive referencing. Complete referencing is impossible, given the large breadth 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 an entry into the field.

Many figures and illustrations have been redone. The use of color illustrations has been deferred in favor of keeping the price of the book down.

Each chapter or section now starts with a short overview of the methods, described without mathematics. This may be useful for getting a feel for the methods, without embarking on all the mathematical details. The overview is followed by a more detailed mathematical description of the method, including some key references that may be consulted for more details. At the end of the chapter or section, some of the pitfalls and the directions of current research are outlined.

Energy units have been converted from kcal/mol to kJ/mol, based on the general opinion 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 include a summary of the most important mathematical techniques used in the book. The goal is to make the book more self-contained, that is relevant mathematical techniques 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 molecular 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 fundamental background.

A rudimentary treatment of periodic systems has been incorporated in Chapters 3 and 14.

A new Chapter 17 has been introduced to describe statistics and QSAR methods.

I have tried to make the book more modular, that is each chapter is more self-contained. This makes it possible to use only selected chapters, for example for a course, but has the drawback of repeating the same things in several chapters, rather than simply cross-referencing.

Although the modularity has been improved, there are unavoidable interdependencies. Chapters 3, 4 and 5 contain the essentials of electronic structure theory, and most would include Chapter 6 describing density functional methods. Chapter 2 contains a description of empirical force field methods, and this is tightly coupled to the simulation methods in Chapter 14, which of course leans on the statistical mechanics in Chapter 13. Chapter 1 on fundamental issues is of a more philosophical nature, and can be skipped. Chapter 16 on mathematical techniques is mainly for those not already familiar 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 from the first edition, where underlining was used, italic text has also been used for emphasizing important points.

A number of people have offered valuable help and criticisms during the updating process. I would especially like to thank S. P. A. Sauer, H. J. Aa. Jensen, E. J. Baerends and 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 this number is so large that only a very tiny fraction will ever be amenable to investigation, the concept of functional groups means that one does not need to evaluate all compounds in a given class to determine their properties. The number of alkanes meeting 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

R. S. Bohacek, C. McMartin and W. C. Guida,

Medicinal Research Reviews

(1), 3–50 (1996).

Preface to the Third Edition

The changes relative to the second edition are as follows:

Numerous misprints and inaccuracies in the second edition have been corrected. Most likely some new ones have been introduced in the process, please check the book website for the most recent correction list and feel free to report possible problems.

http://www.wiley.com/go/jensen/computationalchemistry3

Methodologies and references in each chapter have been updated with new developments published between 2005 and 2015.

Semi-empirical methods have been moved from Chapter 3 to a separate Chapter 7.

Some specific new topics that have been included:

Polarizable force fields

Tight-binding DFT

More extensive DFT functionals, including range-separated and dispersion corrected functionals

More extensive covering of excited states

More extensive time-dependent molecular properties

Accelerated molecular dynamics methods

Tensor decomposition methods

Cluster analysis

Reduced scaling and reduced prefactor methods.

A reoccuring request over the years for a third edition has been: “It would be very useful to have recommendations on which method to use for a given type of problem.” I agree that this would be useful, but I have refrained from it for two main reasons:

Problems range from very narrow ones for a small set of systems, to very broad ones for a wide set of systems, and covering these and all intermediate cases even rudementary is virtually impossible.

Making recommendations like “

do not use method XXX because it gives poor results

” will immediately invoke harsh responses from the developers of method XXX, showing that it gives good results for a selected subset of problems and systems.

A vivid example of the above is the pletora of density functional methods where a particular functional often gives good results for a selected subset of systems and properties, but may fail for other subsets of systems and properties, and no current functional provides good results for all systems and properties. I have limited the recommendations to point out well-known deficiencies.

A similar problem is present when selecting references. I have selected references based on three overriding principles:

References to work containing reference data, such as experimental structural results, or ground-breaking work, such as the Hohenberg–Koch theorem, are to the original work.

Early in each chapter or subsection, I have included review-type papers, where these are available.

Lacking review-type papers, I have selected one or a few papers that preferably are recent, but must at the same time also be written in a scholarly style, and should contain a good selection of references.

The process of literature searching has improved tremendously over the years, and having a few entry points usually allows searching both backwards and forwards to find other references within the selected topic.

In relation to the quoted number of compounds possible for a given number of atoms, Ruddigkeit et al. have estimated the number of plausible compounds composed of H, C, N, O, S and a halogen with up to 17 non-hydrogen atoms to be 166 × 109.1

Reference

L. Ruddigkeit, R. van Deursen, L. C. Blum and J.-L. Reymond,

Journal of Chemical Information and Modeling

(11), 2864–2875 (2012).

1Introduction

Chemistry is the science dealing with construction, transformation and properties of molecules. Theoretical chemistry is the subfield where mathematical methods are combined with fundamental laws of physics to study processes of chemical relevance.1–7

Molecules are traditionally considered as “composed” of atoms or, in a more general sense, as a collection of charged particles, positive nuclei and negative electrons. The only important physical force for chemical phenomena is the Coulomb interaction between these charged particles. Molecules differ because they contain different nuclei and numbers of electrons, or because the nuclear centers are at different geometrical positions. The latter may be “chemically different” molecules such as ethanol and dimethyl ether or different “conformations” of, for example, butane.

Given a set of nuclei and electrons, theoretical chemistry can attempt to calculate things 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 two particles, where the latter can be separated into two pseudo one-particle problems by introducing a “center of mass” coordinate system. Numerical solutions to a given accuracy (which may be so high that the solutions are essentially “exact”) can be generated for many-body systems, by performing a very large number of mathematical operations. Prior to the advent of electronic computers (i.e. before 1950), the number of systems that could be treated with a high accuracy was thus very limited. During the 1960s and 1970s, electronic computers evolved from a few very expensive, difficult 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 of computers is now widespread in many branches of science. This has spawned a new field in chemistry, computational chemistry, where the computer is used as an “experimental” tool, much like, for example, an NMR (nuclear magnetic resonance) spectrometer.

Computational chemistry is focused on obtaining results relevant to chemical problems, not directly at developing new theoretical methods. There is of course a strong interplay between traditional theoretical chemistry and computational chemistry. Developing new theoretical models may enable new problems to be studied, and results from calculations may reveal limitations and suggest improvements in the underlying theory. Depending on the accuracy wanted, and the nature of the system at hand, one can today obtain useful information for systems containing up to several thousand particles. One of the main problems in computational chemistry is selecting a suitable level of theory for a given problem and to be able to evaluate the quality of the obtained results. The present book will try to put the variety of modern computational methods into perspective, hopefully giving the reader a chance of estimating which types of problems can benefit from calculations.

1.1 Fundamental Issues

Before embarking on a detailed description of the theoretical methods in computational chemistry, it may be useful to take a wider look at the background for the theoretical models and how they relate to methods in other parts of science, such as physics and astronomy.

A very large fraction of the computational resources in chemistry and physics is used in 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 than two particles cannot be solved by analytical methods. Computational methods can, however, produce approximate solutions, which in principle may be refined to any desired degree of accuracy.

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

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

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

Branching depending on a decision (add two numbers if

> 10, else subtract one number from the other).

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

Reading and writing data from and to external files.

These tasks are the essence of any programming language, although the syntax, data handling and efficiency depend on the language. The main reason why computers are so useful is the sheer speed with which they can perform these operations. Even a cheap off-the-shelf personal computer can perform billions (109) of operations per second.

Within the scientific world, computers are used for two main tasks: performing numerically intensive calculations and analyzing large amounts of data. The latter can, for example, be pictures generated by astronomical telescopes or gene sequences in the bioinformatics area that need to be compared. The numerically intensive tasks are typically related to simulating the behavior of the real world, by a more or less sophisticated computational model. The main problem in simulations is the multiscale nature of real-world problems, often spanning from subnanometers to millimeters (10−10−10−3) in spatial dimensions and from femtoseconds to milliseconds (10−15−10−3) in the time domain.