Computational Physics

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. 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 "This is an introduction to the physics, astrophysics and cosmology of the cosmic microwave background radiation. The standard big bang model of the universe is adopted at the outset. The topics then covered include the origin of the background, then intrinsic fluctuations, followed by the universe and background radiation after recombination. Finally, measurement of the radiation and its anisotropies is presented, together with a review of the current status of results and experiments. The level is ideally suited to final-year undergraduates in physics or astronomy."--BOOK JACKET. With an emphasis on condensed matter applications, this volume describes computational methods used in theoretical physics. Following an overview of the topic, the book moves on to explore quantum scattering with a spherically symmetric potential Jim Qwilleran Investigates The Murder Of A Chicago Jewelry Dealer As Pickax's Scottish Games Get Underway. J. M. Thijssen. Includes Bibliographical References And Index.