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1 Introduction 1

1.1 Physics and computational physics 1

1.2 Classical mechanics and statistical mechanics 1

1.3 Stochastic simulations 4

1.4 Electrodynamics and hydrodynamics 5

1.5 Quantum mechanics 6

1.6 Relations between quantum mechanics and classical statistical physics 7

1.7 Quantum molecular dynamics 8

1.8 Quantum field theory 9

1.9 About this book 9

Exercises 11

References 13

2 Quantum scattering with a spherically symmetric potential 14

2.1 Introduction 14

2.2 A program for calculating cross sections 18

2.3 Calculation of scattering cross sections 25

Exercises 27

References 28

3 The variational method for the Schr？dinger equation 29

3.1 Variational calculus 29

3.2 Examples of variational calculations 32

3.3 Solution of the generalised eigenvalue problem 36

3.4 Perturbation theory and variational calculus 37

Exercises 39

References 41

4 The Hartree-Fock method 43

4.1 Introduction 43

4.2 The Born-Oppenheimer approximation and the independent-particle method 44

4.3 The helium atom 46

4.4 Many-electron systems and the Slater determinant 52

4.5 Self-consistency and exchange：Hartree-Fock theory 54

4.6 Basis functions 60

4.7 The structure of a Hartree-Fock computer program 69

4.8 Integrals involving Gaussian functions 73

4.9 Applications and results 77

4.10 Improving upon the Hartree-Fock approximation 78

Exercises 80

References 87

5 Density functional theory 89

5.1 Introduction 89

5.2 The local density approximation 95

5.3 Exchange and correlation：a closer look 97

5.4 Beyond DFT：one- and two-particle excitations 101

5.5 A density functional program for the helium atom 109

5.6 Applications and results 114

Exercises 116

References 119

6 Solving the Schr？dinger equation in periodic solids 122

6.1 Introduction：definitions 123

6.2 Band structures and Bloch’s theorem 124

6.3 Approximations 126

6.4 Band structure methods and basis functions 133

6.5 Augmented plane wave methods 135

6.6 The linearised APW（LAPW）method 141

6.7 The pseudopotential method 144

6.8 Extracting information from band structures 160

6.9 Some additional remarks 162

6.10 Other band methods 163

Exercises 163

References 167

7 Classical equilibrium statistical mechanics 169

7.1 Basic theory 169

7.2 Examples of statistical models;phase transitions 176

7.3 Phase transitions 184

7.4 Determination of averages in simulations 192

Exercises 194

References 195

8 Molecular dynamics simulations 197

8.1 Introduction 197

8.2 Molecular dynamics at constant energy 200

8.3 A molecular dynamics simulation program for argon 208

8.4 Integration methods：symplectic integrators 211

8.5 Molecular dynamics methods for different ensembles 223

8.6 Molecular systems 232

8.7 Long-range interactions 241

8.8 Langevin dynamics simulation 247

8.9 Dynamical quantities：nonequilibrium molecular dynamics 251

Exercises 253

References 259

9 Quantum molecular dynamics 263

9.1 Introduction 263

9.2 The molecular dynamics method 266

9.3 An example：quantum molecular dynamics for the hydrogen molecule 272

9.4 Orthonormalisation;conjugate gradient and RM-DIIS techniques 278

9.5 Implementation of the Car-Parrinello technique for pseudopotential DFT 289

Exercises 290

References 293

10 The Monte Carlo method 295

10.1 Introduction 295

10.2 Monte Carlo integration 296

10.3 Importance sampling through Markov chains 299

10.4 Other ensembles 310

10.5 Estimation of free energy and chemical potential 316

10.6 Further applications and Monte Carlo methods 319

10.7 The temperature of a finite system 330

Exercises 334

References 335

11 Transfer matrix and diagonalisation of spin chains 338

11.1 Introduction 338

11.2 The one-dimensional Ising model and the transfer matrix 339

11.3 Two-dimensional spin models 343

11.4 More complicated models 347

11.5 ’Exact’diagonalisation of quantum chains 349

11.6 Quantum renormalisation in real space 355

11.7 The density matrix renormalisation group method 358

Exercises 365

References 370

12 Quantum Monte Carlo methods 372

12.1 Introduction 372

12.2 The variational Monte Carlo method 373

12.3 Diffusion Monte Carlo 387

12.4 Path-integral Monte Carlo 398

12.5 Quantum Monte Carlo on a lattice 410

12.6 The Monte Carlo transfer matrix method 414

Exercises 417

References 421

13 The finite element method for partial differential equations 423

13.1 Introduction 423

13.2 The Poisson equation 424

13.3 Linear elasticity 429

13.4 Error estimators 434

13.5 Local refinement 436

13.6 Dynamical finite element method 439

13.7 Concurrent coupling of length scales：FEM and MD 440

Exercises 445

References 446

14 The lattice Boltzmann method for fluid dynamics 448

14.1 Introduction 448

14.2 Derivation of the Navier-Stokes equations 449

14.3 The lattice Boltzmann model 455

14.4 Additional remarks 458

14.5 Derivation of the Navier-Stokes equation from the lattice Boltzmann model 460

Exercises 463

References 464

15 Computational methods for lattice field theories 466

15.1 Introduction 466

15.2 Quantum field theory 467

15.3 Interacting fields and renormalisation 473

15.4 Algorithms for lattice field theories 477

15.5 Reducing critical slowing down 491

15.6 Comparison of algorithms for scalar field theory 509

15.7 Gauge field theories 510

Exercises 532

References 536

16 High performance computing and parallelism 540

16.1 Introduction 540

16.2 Pipelining 541

16.3 Parallelism 545

16.4 Parallel algorithms for molecular dynamics 552

References 556

Appendix A Numerical methods 557

A1 About numerical methods 557

A2 Iterative procedures for special functions 558

A3 Finding the root of a function 559

A4 Finding the optimum of a function 560

A5 Discretisation 565

A6 Numerical quadratures 566

A7 Differential equations 568

A8 Linear algebra problems 590

A9 The fast Fourier transform 598

Exercises 601

References 603

Appendix B Random number generators 605

B1 Random numbers and pseudo-random numbers 605

B2 Random number generators and properties of pseudo-random numbers 606

B3 Nonuniform random number generators 609

Exercises 611

References 612

Index 613