Computational Physics

✍ Scribed by J. M. Thijssen

• Publisher: Cambridge University Press
• Year: 1999
• Tongue: English
• Leaves: 559
• Category: Library

✦ Synopsis

Computional physics involves the use of computer calculations and simulations to solve physical problems. This book describes computational methods used in theoretical physics with emphasis on condensed matter applications. Coverage begins with an overview of the wide variety of topics and algorithmic approaches studied in this book. The next chapters concentrate on electronic structure calculations, presenting the Hartree-Fock and Density Functional formalisms, and band structure methods. Later chapters discuss molecular dynamics simulations and Monte Carlo methods in classical and quantum physics, with applications to condensed matter and particle field theories. Each chapter details the necessary fundamentals, describes the formation of a sample program, and includes problems that address related analytical and numerical issues. Useful appendices on numerical methods and random number generators are also included. This volume bridges the gap between undergraduate physics and computational research. It is an ideal textbook for graduate students as well as a valuable reference for researchers.

✦ Table of Contents

Contents......Page 4
Preface......Page 10
1.1 Physics and computational physics......Page 14
1.2 Classical mechanics and statistical mechanics......Page 15
1.3 Stochastic simulations......Page 17
1.5 Quantum mechanics......Page 19
1.6 Relations between quantum mechanics and classical statistical physics......Page 21
1.8 Quantum field theory......Page 22
1.9 About this book......Page 23
2.1 Introduction......Page 27
2.2.1 Numerov's algorithm for the radial Schrodinger equation......Page 32
2.2.2 The spherical Bessel functions......Page 35
2.2.3 Putting the pieces together - results......Page 36
2.3 Calculation of scattering cross sections......Page 37
3.1 Variational calculus......Page 43
3.2 Examples of variational calculations......Page 46
3.2.1 The infinitely deep potential well......Page 47
3.2.2 Variational calculation for the hydrogen atom......Page 48
3.3 Solution of the generalised eigenvalue problem......Page 51
3.4 Perturbation theory and variational calculus......Page 52
4.1 Introduction......Page 58
4.2 The Born-Oppenheimer approximation and the IP method......Page 59
4.3.1 Self-consistency......Page 61
4.3.2 A program for calculating the helium ground state......Page 64
4.4 Many-electron systems and the Slater determinant......Page 67
4.5.1 The Hartree-Fock equations - physical picture......Page 70
4.5.2 Derivation of the Hartree-Fock equations......Page 72
4.6 Basis functions......Page 76
4.6.1 Closed- and open-shell systems......Page 77
4.6.2 Basis functions: STO and GTO......Page 80
4.7 The structure of a Hartree-Fock computer program......Page 85
4.7.1 The two-electron integrals......Page 86
4.7.2 General scheme of the HF program......Page 87
4.8 Integrals involving Gaussian functions......Page 90
4.9 Applications and results......Page 95
4.10 Improving upon the Hartree-Fock approximation......Page 96
5.1 Introduction......Page 107
5.1.1 Density functional theory - physical picture......Page 108
5.1.2 Density functional formalism and derivation of the Kohn-Sham equations......Page 109
5.2 The local density approximation......Page 113
5.3 A density functional program for the helium atom......Page 115
5.3.1 Solving the radial equation......Page 116
5.3.2 Including the Hartree potential......Page 117
5.3.3 The local density exchange potential......Page 119
5.4 Applications and results......Page 121
6 Solving the Schrodinger equation in periodic solids......Page 127
6.1.2 Reciprocal lattice......Page 128
6.2 Band structures and Bloch's theorem......Page 129
6.3.1 The nearly free electron approximation......Page 131
6.3.2 The tight-binding approximation......Page 132
6.4 Band structure methods and basis functions......Page 134
6.5.1 Plane waves and augmentation......Page 136
6.5.2 An APW program for the band structure of copper......Page 140
6.6 The linearised APW (LAPW) method......Page 143
6.7 The pseudopotential method......Page 146
6.7.1 A pseudopotential band structure program for silicon......Page 148
6.7.2 Accurate energy-independent pseudopotentials......Page 150
6.8 Extracting information from band structures......Page 151
6.9 Some additional remarks......Page 152
6.10 Other band methods......Page 153
7.1 Basic theory......Page 159
7.1.1 Ensembles......Page 160
7.2.1 Molecular systems......Page 167
7.2.2 Lattice models......Page 170
7.3.1 First order and continuous phase transitions......Page 175
7.3.2 Critical phase transitions and finite size scaling......Page 177
7.4 Determination of averages in simulations......Page 184
8.1 Introduction......Page 188
8.2 Molecular dynamics at constant energy......Page 192
8.3 A molecular dynamics simulation program for argon......Page 198
8.4 Integration methods - symplectic integrators......Page 201
8.4.1 The Verlet algorithm revisited......Page 202
8.4.2 Symplectic geometry - symplectic integrators......Page 208
8.4.3 Derivation of symplectic integrators......Page 211
8.5.1 Constant temperature......Page 215
8.5.2 Keeping the pressure constant......Page 222
8.6.1 Molecular degrees of freedom......Page 224
8.6.2 Rigid molecules......Page 225
8.6.3 General procedure - partial constraints......Page 231
8.7 Long range interactions......Page 233
8.7.1 The periodic Coulomb interaction......Page 234
8.7.2 Efficient evaluation of forces and potentials......Page 236
8.8 Langevin dynamics simulation......Page 240
8.9 Dynamical quantities - nonequilibrium molecular dynamics......Page 244
9.1 Introduction......Page 255
9.2 The molecular dynamics method......Page 259
9.3 An example: quantum molecular dynamics for the hydrogen molecule......Page 264
9.3.1 The electronic structure......Page 265
9.3.2 The nuclear motion......Page 266
9.4 Orthonormalisation; conjugate gradient techniques......Page 271
9.4.1 Orthogonalisation of the electronic orbitals......Page 272
9.4.2 The conjugate gradient method......Page 276
9.4.3 Large systems......Page 279
10.1 Introduction......Page 284
10.2 Monte Carlo integration......Page 285
10.3 Importance sampling through Markov chains......Page 288
10.3.1 Monte Carlo for the Ising model......Page 293
10.3.2 Monte Carlo simulation of a monatomic gas......Page 298
10.4.1 The (NPT) ensemble......Page 300
10.4.2 The grand canonical ensemble......Page 302
10.4.3 The Gibbs ensemble......Page 304
10.5 Estimation of free energy and chemical potential......Page 306
10.5.1 Free energy calculation......Page 307
10.5.2 Chemical potential determination......Page 309
11.1 Introduction......Page 312
11.2 The one-dimensional Ising model and the transfer matrix......Page 313
11.3 Two-dimensional spin models......Page 317
11.4 More complicated models......Page 321
12.1 Introduction......Page 326
12.2.1 Description of the method......Page 327
12.2.2 Sample programs and results......Page 329
12.2.3 Trial functions......Page 331
12.2.4 Diffusion equations, Green's functions and Langevin equations......Page 333
12.2.5 The Fokker-Planck equation approach to VMC......Page 339
12.3.1 Simple diffusion Monte Carlo......Page 341
12.3.2 Applications......Page 344
12.3.3 Guide function for diffusion Monte Carlo......Page 345
12.3.4 Problems with fermion calculations......Page 348
12.4 Path integral Monte Carlo......Page 352
12.4.1 Path integral fundamentals......Page 353
12.4.2 Applications......Page 361
12.4.3 Increasing the efficiency......Page 364
12.5 Quantum Monte Carlo on a lattice......Page 366
12.6 The Monte Carlo transfer matrix method......Page 370
13.1 Introduction......Page 378
13.2 Quantum field theory......Page 379
13.3 Interacting fields and renormalisation......Page 386
13.4 Algorithms for lattice field theories......Page 390
13.4.1 Monte Carlo methods......Page 393
13.4.2 The MC algorithms: implementation and results......Page 395
13.4.3 Molecular dynamics......Page 398
13.5 Reducing critical slowing down......Page 405
13.5.1 The Swendsen-Wang method......Page 406
13.5.2 Wolff's single cluster algorithm......Page 410
13.5.3 The multigrid Monte Carlo method......Page 417
13.5.4 The Fourier-accelerated Langevin method......Page 420
13.6 Comparison of algorithms for scalar field theory......Page 422
13.7.1 The electromagnetic Lagrangian......Page 423
13.7.2 Electromagnetism on a lattice - quenched compact QED......Page 428
13.7.3 A lattice QED simulation......Page 433
13.7.4 Including dynamical fermions......Page 436
13.7.5 Nonabelian gauge fields - quantum chromodynamics......Page 443
14.1 Introduction......Page 454
14.2.1 Architectural aspects......Page 455
14.2.2 Implications for programming......Page 458
14.3.1 Parallel architectures......Page 460
14.3.2 Programming implications......Page 463
14.4 A systolic algorithm for molecular dynamics......Page 467
A.1 About numerical methods......Page 472
A.2 Iterative procedures for special functions......Page 473
A.3 Finding the root of a function......Page 474
A.4 Finding the optimum of a function......Page 476
A.5 Discretisation......Page 481
A.6 Numerical quadrature......Page 482
A.7 Differential equations......Page 484
A.7.1 Ordinary differential equations......Page 485
A.7.2 Partial differential equations......Page 494
A.8.1 Systems of linear equations......Page 506
A.8.2 Matrix diagonalisation......Page 511
A.9.1 General considerations......Page 515
A.9.2 How does the FFT work?......Page 517
B.1 Random numbers and pseudo-random numbers......Page 522
B.2 Random number generators and properties of pseudorandom numbers......Page 523
B.3 Nonuniform random number generators......Page 526
References......Page 531
Index......Page 551

✦ Subjects

Физика;Матметоды и моделирование в физике;