Computational Physics. J. M. Thijssen

Transcription

1 Computational Physics J. M. Thijssen Delft University of Technology CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface xi 1 Introduction Physics and computational physics Classical mechanics and statistical mechanics Stochastic simulations Electrodynamics and hydrodynamics Quantum mechanics Relations between quantum mechanics and classical statistical physics Quantum molecular dynamics Quantum field theory About this book 10 2 Quantum scattering with a spherically symmetric potential Introduction A program for calculating cross sections Numerov's algorithm for the radial Schrodinger equation The spherical Bessel functions Putting the pieces together - results Calculation of scattering cross sections 24 3 The variational method for the Schrodinger equation Variational calculus Examples of variational calculations The infinitely deep potential well Variational calculation for the hydrogen atom Solution of the generalised eigenvalue problem Perturbation theory and variational calculus 39

3 vi Contents 4 The Hartree-Fock method Introduction The Born-Oppenheimer approximation and the IP method The helium atom Self-consistency A program for calculating the helium ground state Many-electron systems and the Slater determinant Self-consistency and exchange: Hartree-Fock theory The Hartree-Fock equations - physical picture Derivation of the Hartree-Fock equations Koopman's theorem Basis functions Closed- and open-shell systems Basis functions: STO and GTO The structure of a Hartree-Fock computer program The two-electron integrals General scheme of the HF program Integrals involving Gaussian functions Applications and results Improving upon the Hartree-Fock approximation 83 5 Density functional theory Introduction Density functional theory - physical picture Density functional formalism and derivation of the Kohn-Sham equations The local density approximation A density functional program for the helium atom Solving the radial equation Including the Hartree potential The local density exchange potential Applications and results Solving the Schrodinger equation in periodic solids Introduction - definitions Crystal lattices Reciprocal lattice Band structures and Bloch's theorem Approximations The nearly free electron approximation 118

4 Contents vii The tight-binding approximation Band structure methods and basis functions Augmented plane wave methods Plane waves and augmentation An APW program for the band structure of copper The linearised APW (LAPW) method The pseudopotential method A pseudopotential band structure program for silicon Accurate energy-independent pseudopotentials Extracting information from band structures Some additional remarks Other band methods Classical statistical mechanics Basic theory Ensembles Examples of statistical models - phase transitions Molecular systems Lattice models Phase transitions First order and continuous phase transitions Critical phase transitions and finite size scaling Determination of averages in simulations Molecular dynamics simulations Introduction Molecular dynamics at constant energy A molecular dynamics simulation program for argon Integration methods - symplectic integrators The Verlet algorithm revisited Symplectic geometry - symplectic integrators Derivation of symplectic integrators Molecular dynamics methods for different ensembles Constant temperature Keeping the pressure constant Molecular systems Molecular degrees of freedom Rigid molecules General procedure - partial constraints Long range interactions 220

5 viii Contents The periodic Coulomb interaction Efficient evaluation of forces and potentials Langevin dynamics simulation Dynamical quantities - nonequilibrium molecular dynamics Quantum molecular dynamics Introduction The molecular dynamics method An example: quantum molecular dynamics for the hydrogen molecule The electronic structure The nuclear motion Orthonormalisation; conjugate gradient techniques Orthogonalisation of the electronic orbitals The conjugate gradient method Large systems The Monte Carlo method Introduction Monte Carlo integration Importance sampling through Markov chains Monte Carlo for the Ising model Monte Carlo simulation of a monatomic gas Other ensembles The (NPT) ensemble The grand canonical ensemble The Gibbs ensemble Estimation of free energy and chemical potential Free energy calculation Chemical potential determination Transfer matrix methods Introduction The one-dimensional Ising model and the transfer matrix Two-dimensional spin models More complicated models Quantum Monte Carlo methods Introduction The variational Monte Carlo method 314

6 Contents ix Description of the method Sample programs and results Trial functions Diffusion equations, Green's functions and Langevin equations The Fokker-Planck equation approach to VMC Diffusion Monte Carlo Simple diffusion Monte Carlo Applications Guide function for diffusion Monte Carlo Problems with fermion calculations Path integral Monte Carlo Path integral fundamentals Applications Increasing the efficiency Quantum Monte Carlo on a lattice The Monte Carlo transfer matrix method Computational methods for lattice field theories Introduction Quantum field theory Interacting fields and renormalisation Algorithms for lattice field theories Monte Carlo methods The MC algorithms: implementation and results Molecular dynamics Reducing critical slowing down The Swendsen-Wang method Wolff's single cluster algorithm The multigrid Monte Carlo method The Fourier-accelerated Langevin method Comparison of algorithms for scalar field theory Gauge field theories The electromagnetic Lagrangian Electromagnetism on a lattice - quenched compact QED A lattice QED simulation Including dynamical fermions Nonabelian gauge fields - quantum chromodynamics. 430

7 x Contents 14 High performance computing and parallelism Introduction Pipelining Architectural aspects Implications for programming Parallelism Parallel architectures Programming implications A systolic algorithm for molecular dynamics 454 Appendix A Numerical methods 459 A. 1 About numerical methods 459 A.2 Iterative procedures for special functions 460 A.3 Finding the root of a function 461 A.4 Finding the optimum of a function 463 A.5 Discretisation 468 A.6 Numerical quadrature 469 A.7 Differential equations 471 A.7.1 Ordinary differential equations 472 A.7.2 Partial differential equations 481 A.8 Linear algebra problems 493 A.8.1 Systems of linear equations 493 A.8.2 Matrix diagonalisation 498 A.9 The fast Fourier transform 502 A.9.1 General considerations 502 A.9.2 How does the FFT work? 504 Appendix B Random number generators 509 B.I Random numbers and pseudo-random numbers 509 B.2 Random number generators and properties of pseudorandom numbers 510 B.3 Nonuniform random number generators 513 References 518 Index 538