more information – www.cambridge.org/9780521833462 ComputationalPhysics SecondEdition This Second Edition has been fully updated. The wide range of topics covered in the First Edition has been extended with new chapters on finite element methods andlatticeBoltzmannsimulation.Newsectionshavebeenaddedtothechapterson density functional theory, quantum molecular dynamics, Monte Carlo simulation anddiagonalisationofone-dimensionalquantumsystems. The book covers many different areas of physics research and different computa- tionalmethodologies,withanemphasisoncondensedmatterphysicsandphysical chemistry.ItincludescomputationalmethodssuchasMonteCarloandmolecular dynamics, various electronic structure methodologies, methods for solving par- tial differential equations, and lattice gauge theory. Throughout the book, the relations between the methods used in different fields of physics are emphas- ised. Several new programs are described and these can be downloaded from www.cambridge.org/9780521833462 The book requires a background in elementary programming, numerical analysis and field theory, as well as undergraduate knowledge of condensed matter theory andstatisticalphysics.Itwillbeofinteresttograduatestudentsandresearchersin theoretical,computationalandexperimentalphysics. JosThijssenisalecturerattheKavliInstituteofNanoscienceatDelftUniversity ofTechnology. COMPUTATIONAL PHYSICS Second Edition JOSTHIJSSEN KavliInstituteofNanoscience,DelftUniversityofTechnology CAMBRIDGEUNIVERSITYPRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo Delhi,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521833462 ©J.M.Thijssen1999,2007 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished1999 Reprinted2003 Secondedition2007 3rdprinting2012 Printedand bound by MPG Books Group, UK AcataloguerecordforthispublicationisavailablefromtheBritishLibrary ISBN-13978-0-521-83346-2Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurlsforexternalor third-partyinternetwebsitesreferredtointhispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Prefacetothefirstedition pagexi Prefacetothesecondedition xiv 1 Introduction 1 1.1 Physicsandcomputationalphysics 1 1.2 Classicalmechanicsandstatisticalmechanics 1 1.3 Stochasticsimulations 4 1.4 Electrodynamicsandhydrodynamics 5 1.5 Quantummechanics 6 1.6 Relationsbetweenquantummechanicsandclassical statisticalphysics 7 1.7 Quantummoleculardynamics 8 1.8 Quantumfieldtheory 9 1.9 Aboutthisbook 9 Exercises 11 References 13 2 Quantumscatteringwithasphericallysymmetric potential 14 2.1 Introduction 14 2.2 Aprogramforcalculatingcrosssections 18 2.3 Calculationofscatteringcrosssections 25 Exercises 27 References 28 3 ThevariationalmethodfortheSchrödingerequation 29 3.1 Variationalcalculus 29 3.2 Examplesofvariationalcalculations 32 3.3 Solutionofthegeneralisedeigenvalueproblem 36 3.4 Perturbationtheoryandvariationalcalculus 37 v vi Contents Exercises 39 References 41 4 TheHartree–Fockmethod 43 4.1 Introduction 43 4.2 TheBorn–Oppenheimerapproximationandthe independent-particlemethod 44 4.3 Theheliumatom 46 4.4 Many-electronsystemsandtheSlaterdeterminant 52 4.5 Self-consistencyandexchange:Hartree–Focktheory 54 4.6 Basisfunctions 60 4.7 ThestructureofaHartree–Fockcomputerprogram 69 4.8 IntegralsinvolvingGaussianfunctions 73 4.9 Applicationsandresults 77 4.10 ImprovingupontheHartree–Fockapproximation 78 Exercises 80 References 87 5 Densityfunctionaltheory 89 5.1 Introduction 89 5.2 Thelocaldensityapproximation 95 5.3 Exchangeandcorrelation:acloserlook 97 5.4 BeyondDFT:one-andtwo-particleexcitations 101 5.5 Adensityfunctionalprogramfortheheliumatom 109 5.6 Applicationsandresults 114 Exercises 116 References 119 6 SolvingtheSchrödingerequationinperiodicsolids 122 6.1 Introduction:definitions 123 6.2 BandstructuresandBloch’stheorem 124 6.3 Approximations 126 6.4 Bandstructuremethodsandbasisfunctions 133 6.5 Augmentedplanewavemethods 135 6.6 ThelinearisedAPW(LAPW)method 141 6.7 Thepseudopotentialmethod 144 6.8 Extractinginformationfrombandstructures 160 6.9 Someadditionalremarks 162 6.10 Otherbandmethods 163 Contents vii Exercises 163 References 167 7 Classicalequilibriumstatisticalmechanics 169 7.1 Basictheory 169 7.2 Examplesofstatisticalmodels;phasetransitions 176 7.3 Phasetransitions 184 7.4 Determinationofaveragesinsimulations 192 Exercises 194 References 195 8 Moleculardynamicssimulations 197 8.1 Introduction 197 8.2 Moleculardynamicsatconstantenergy 200 8.3 Amoleculardynamicssimulationprogramforargon 208 8.4 Integrationmethods:symplecticintegrators 211 8.5 Moleculardynamicsmethodsfordifferentensembles 223 8.6 Molecularsystems 232 8.7 Long-rangeinteractions 241 8.8 Langevindynamicssimulation 247 8.9 Dynamicalquantities:nonequilibriummoleculardynamics 251 Exercises 253 References 259 9 Quantummoleculardynamics 263 9.1 Introduction 263 9.2 Themoleculardynamicsmethod 266 9.3 Anexample:quantummoleculardynamicsforthehydrogen molecule 272 9.4 Orthonormalisation;conjugategradientandRM-DIIS techniques 278 9.5 ImplementationoftheCar–Parrinellotechniquefor pseudopotentialDFT 289 Exercises 290 References 293 10 TheMonteCarlomethod 295 10.1 Introduction 295 10.2 MonteCarlointegration 296 10.3 ImportancesamplingthroughMarkovchains 299 viii Contents 10.4 Otherensembles 310 10.5 Estimationoffreeenergyandchemicalpotential 316 10.6 FurtherapplicationsandMonteCarlomethods 319 10.7 Thetemperatureofafinitesystem 330 Exercises 334 References 335 11 Transfermatrixanddiagonalisationofspinchains 338 11.1 Introduction 338 11.2 Theone-dimensionalIsingmodelandthe transfermatrix 339 11.3 Two-dimensionalspinmodels 343 11.4 Morecomplicatedmodels 347 11.5 ‘Exact’diagonalisationofquantumchains 349 11.6 Quantumrenormalisationinrealspace 355 11.7 Thedensitymatrixrenormalisationgroupmethod 358 Exercises 365 References 370 12 QuantumMonteCarlomethods 372 12.1 Introduction 372 12.2 ThevariationalMonteCarlomethod 373 12.3 DiffusionMonteCarlo 387 12.4 Path-integralMonteCarlo 398 12.5 QuantumMonteCarloonalattice 410 12.6 TheMonteCarlotransfermatrixmethod 414 Exercises 417 References 421 13 Thefiniteelementmethodforpartialdifferentialequations 423 13.1 Introduction 423 13.2 ThePoissonequation 424 13.3 Linearelasticity 429 13.4 Errorestimators 434 13.5 Localrefinement 436 13.6 Dynamicalfiniteelementmethod 439 13.7 Concurrentcouplingoflengthscales:FEMandMD 440 Exercises 445 References 446