WolfgangSchattkeand RicardoDíezMuiño QuantumMonteCarlo Programming RelatedTitles Waser,R.(ed.) Allinger,N.L. Nanoelectronicsand Molecular Structure InformationTechnology UnderstandingStericandElectronic AdvancedElectronicMaterialsand EffectsfromMolecularMechanics NovelDevices 2010 ISBN:978-0-470-19557-4 2012 ISBN:978-3-527-40927-3 Owens,F.J.,Poole,Jr.,C.P. Reimers,J.R. The Physicsand Chemistryof Computational Methods for Nanosolids LargeSystems 2008 ElectronicStructureApproachesfor ISBN:978-0-470-06740-6 BiotechnologyandNanotechnology 2011 ISBN:978-0-470-48788-4 Reinhard,P.-G.,Suraud,E. Introduction toCluster Dynamics Alkauskas,A.,Deák,P.,Neugebauer,J., Pasquarello,A.,VandeWalle,C.G.(eds.) 2004 Advanced Calculations for ISBN:978-3-527-40345-5 DefectsinMaterials ElectronicStructureMethods Landau,R.H.,Páez,M.J. 2011 Computational Physics ISBN:978-3-527-41024-8 ProblemSolvingwithComputers 1997 Kroese,D.P.,Taimre,T.,Botev,Z.I. ISBN:978-0-471-11590-8 Handbook of Monte Carlo Methods 2011 ISBN:978-0-470-17793-8 Wolfgang Schattke and Ricardo Díez Muiño Quantum Monte Carlo Programming for Atoms,Molecules,Clusters,and Solids TheAuthors AllbookspublishedbyWiley-VCHarecarefully produced.Nevertheless,authors,editors,and Prof.WolfgangSchattke publisherdonotwarranttheinformation InstituteofTheoreticalPhysicsandAstrophysics containedinthesebooks,includingthisbook,to Christian-Albrechts-UniversityKiel befreeoferrors.Readersareadvisedtokeepin Leibnizstr.15 mindthatstatements,data,illustrations, 24118Kiel proceduraldetailsorotheritemsmay inadvertentlybeinaccurate. and LibraryofCongressCardNo.: IkerbasqueFoundation/DonostiaInternational appliedfor PhysicsCenter BritishLibraryCataloguing-in-PublicationData: P.ManueldeLardizabal4 Acataloguerecordforthisbookisavailable 20018Donostia–SanSebastián fromtheBritishLibrary. Spain Bibliographicinformationpublishedbythe Dr.RicardoDíezMuiño DeutscheNationalbibliothek CentrodeFísicadeMaterialesCSIC-UPV/EHU TheDeutscheNationalbibliothekliststhis and publicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableonthe DonostiaIntern.PhysicsCente Internetathttp://dnb.d-nb.de. P.ManueldeLardizabal4 20018Donostia–SanSebastian ©2013WILEY-VCHVerlagGmbH&Co.KGaA, Spain Boschstr.12,69469Weinheim,Germany Allrightsreserved(includingthoseoftranslation intootherlanguages).Nopartofthisbookmay bereproducedinanyform–byphotoprinting, microfilm,oranyothermeans–nortransmitted ortranslatedintoamachinelanguagewithout writtenpermissionfromthepublishers.Regis- terednames,trademarks,etc.usedinthisbook, evenwhennotspecificallymarkedassuch,are nottobeconsideredunprotectedbylaw. PrintISBN 978-3-527-40851-1 ePDFISBN 978-3-527-67574-6 ePubISBN 978-3-527-67532-6 mobiISBN 978-3-527-67531-9 Composition le-texpublishingservicesGmbH, Leipzig PrintingandBinding MarkonoPrintMedia PteLtd,Singapore CoverDesign Adam-Design,Weinheim PrintedinSingapore Printedonacid-freepaper V Contents Preface IX 1 AFirstMonteCarloExample 1 1.1 EnergyofInteractingClassicalGas 1 1.1.1 ClassicalMany-ParticleStatisticsandSomeThermodynamics 2 1.1.2 HowtoSampletheParticleDensity? 18 2 VariationalQuantumMonteCarloforaOne-ElectronSystem 23 3 Two Electrons with Two Adiabatically Decoupled Nuclei: HydrogenMolecule 39 3.1 TheoreticalDescriptionoftheSystem 39 3.2 NumericalResultsofModerateAccuracy 42 3.3 ControllingtheAccuracy 46 3.4 DetailsofNumericalProgram 53 4 ThreeElectrons:LithiumAtom 61 4.1 MoreElectrons,MoreProblems:ParticleandSpinSymmetry 63 4.1.1 AntisymmetryandDecompositionoftheMany-BodyWaveFunction 63 4.1.2 Three-ElectronWaveFunction 65 4.1.3 GeneralWaveFunction 67 4.1.4 RelaxingSymmetryofTotalSpin 70 4.2 ElectronOrbitalsfortheSlaterDeterminant 71 4.3 SlaterDeterminants:EvaluationandUpdate 76 4.4 SomeImportantObservablesinAtoms? 82 4.4.1 TheModule“observables” 87 4.5 StatisticalAccuracy 91 4.6 GroundStateResults 93 4.6.1 ResultsforLithiumAtom 93 4.6.2 CodeofMainProgram,ModulesofVariables,ofStatistic,ofJastrow Factor,andofOutput 103 4.7 Optimization? 115 5 Many-ElectronConfinedSystems 121 5.1 ModelSystemswithFewElectrons 121 VI Contents 5.2 OrthorhombicQuantumDot 122 5.2.1 ConfinedSingle-ParticleWaveFunctions 122 5.2.2 DetailsofProgram 123 5.2.3 EnergyandRadialDensity 125 5.2.4 Pair-CorrelationFunction 131 5.2.5 ProgramofthePair-CorrelationFunction 134 5.3 SphericalQuantumDot 136 5.3.1 FundamentalsofDFT 137 5.3.2 DFTCalculationoftheJelliumCluster:Methodology 138 5.3.3 QMCCalculationoftheJelliumCluster:Methodology 140 5.3.4 QMCCodefortheCalculationofJelliumClusters 141 5.3.5 Comparison between DFT and QMC Calculations of Jellium Clusters 142 6 Many-ElectronAtomicAggregates:LithiumCluster 147 6.1 ClustersandNanophysics 147 6.2 CubicBCCArrangementofLithiumAtoms 150 6.2.1 StructureoftheMainProgram 150 6.2.2 Single-ElectronWaveFunctionsandStructureoftheDeterminant 150 6.2.3 GeometricSettingoftheCluster 153 6.2.4 ChangesintheProgram 156 6.3 TheCluster:IntermediatebetweenAtomandSolid 163 6.3.1 1(cid:2)1(cid:2)1Cluster:Li 164 2 6.3.2 2(cid:2)2(cid:2)2Cluster 167 6.3.3 3(cid:2)3(cid:2)3Cluster 172 6.3.4 4(cid:2)4(cid:2)4Cluster 174 6.3.5 ClusterSize 178 7 InfiniteNumberofElectrons:LithiumSolid 181 7.1 InfiniteLattice 183 7.1.1 TheLattices 183 7.1.2 StructureoftheElectrostaticPotential 186 7.1.3 EwaldSummationandTabulation 191 7.1.4 Finite-SizeEffects 204 7.2 WaveFunction 208 7.2.1 LinearCombinationofAtomicOrbitals 208 7.2.2 PlaneWaves 210 7.3 JastrowFactor 212 7.3.1 StandardChoice 213 7.3.2 PrincipalIdeasandExtensions 215 7.4 Resultsforthe3(cid:2)3(cid:2)3and4(cid:2)4(cid:2)4SuperlatticeSolid 216 8 DiffusionQuantumMonteCarlo(DQMC) 223 8.1 TowardsaFirstDQMCProgram 224 8.1.1 RelatingSchrödingerEquationtoDiffusion 224 8.1.2 GenerateGaussianRandomNumbers 228 Contents VII 8.1.3 Application 229 8.1.3.1 HarmonicOscillator 229 8.2 Conclusion 235 9 Epilogue 237 Appendix 239 A.1 TheInteractingClassicalGas:HighTemperatureAsymptotics 239 A.2 PseudorandomNumberGenerators 241 A.3 SomeGeneralizationoftheJastrowFactor 247 A.4 SeriesExpansion 249 A.5 WaveFunctionSymmetryandSpin 257 A.5.1 FourElectrons 257 A.6 InfiniteLattice:EwaldSummation 259 A.7 LatticeSums:Calculation 263 References 269 Index 273 IX Preface Thereadermightbeinclinednottoreadtheprefacewhenstartingwiththebook, butrather atalater timewhenlazinessorleisureleavestimeforit.Intheworst case, thereader might come back to the preface angered by some lack of under- standingor,quitetheopposite,angeredbyreadingsomeundergraduatesimplistic explanations.Byconsultingthepreface,thereaderisaskingtheauthorsabouttheir goalsinwritingthebook. Themaingoalisdeclaredbythebook’stitle,neverthelesswithsomerestrictions inmind. Thefollowingpublicationissettledsomewherebetween atextbookandacom- putercodemanual.Itslevelisperhapstoospecializedforatextbookandtoobroad foramanual.Apositivecommentwouldbethatitscontentincludesratherprac- tical adviceon what isusuallydescribed in atheoretical textbook, as well as pre- senting in more detail the physical understandingof what the manual of a code promisesasaresult.Danglingbetweenthesetwoextremestheauthorscouldnot decidewheretoplacethebookexactly,sotheydecidedtotaketheriskofsharing thecommongroundinboth. Of course, one purpose was to make it more reader-friendly than a scientific paper,orareviewarticle.However,reviewssuchasthatofFoulkes,Mitas,Needs, andRajagopalforexample,represent invaluablesourcesforextendedstudies[1]. Thepathtofulfillingthepurposeofa“friendly”bookwasledonlybytheauthors’ own experience and will differ from that of others. In other words, neither the authors attended courses on “How to Write Pedagogically Good Books” nor did theyreadsuchliterature. Of course, the reader does not expect a manual coming with a scientific code which reduces to “read the input file explanations and then go on.” Therefore, insteadofpresentingjustonecodethatcouldcoverthegeneralfield,theauthors decidedtobreakuptheprogramintopieces,eachofthemdevotedtooneofafew leadingexamples. Apedagogic,buttimeandspace-consumingpossibilitywouldhavebeentode- velopthecodesstepbystep,toletthereaderrunintothemanytrapsofprogram- mingerrorswithexercisesandsolutions.Thatcouldfillmanyvolumes.Wetried nottoexpandthevolumebeyondacceptablelimits,buttokeepenoughmaterialso X Preface thatthereader couldstart anddevelopfromithisorherownspecificprograms. Therefore,wegaveuponthetextbookidea. Attheveryfirstconceptofthisbookwethoughtofpresentingthecodecollected fromthePhDthesesofEcksteinandBahnsen,whocompletedtheirworkwithin thegroupof oneof theauthors(WS). Itsoon became clear that wewould easily runintooneofthedifficultiescitedabovewhichwewantedtoavoid.Therefore,we decidedtoprogramfromscratch.Inthisway,wewerealsofreetopresentourway ofunderstandingthecodes.Inaddition,wetakeonfullresponsibilityoferrors,not attributingthemtoanyothersource. However,thenumberofmistakesunveiledandadditionallythosestillhiddenis embarrassing.Thoughsomeofthelattermightbeusefultotrackthepaththecode developed,theyarenotonpurpose,weassurethat.Wepresentthecodeasitdevel- opedaftertestingandcorrectingasusual.Ourmainprogrammingstyle,ifwehad any,wastorenderthecodetobeeasilychanged.Thiscanbetakenasanexcusefor thelackofbeautyandthelackofprogramefficiency.Bothaspectsandperspectives willbeevaluatedbythecommunitydifferentlywithchangingtime,changingcom- pilers, and changing computationalfacilities. To keep the work along the course offindingthepleasureinwriting,wemustadmitdeficiencieswhichwearenow blamed for. We hope that the pleasure of eventually acquiring successful access tothequantumMonteCarloschememightoutweightheshortcomingsfromthe reader’spointofviewaswell. Thus,thebookisnotwrittentodeliveranoptimizedprogramcode.Thesecodes existandtheirdevelopmentislefttoanotherbranchofscience.Instead,wewanted toshowsomeaspectsofthevastandbeautifulpossibilitiesofthequantumMonte Carlo (QMC) method and to attract and maybe seduce the reader to devote his orherinteresttothissubject.Wealsowanttotouchonthevariouspossibilitiesof choicesofcomputingschemesconnectedwiththemethod.Thematerialpresented hereisbynowaycomplete,andthegeneralscientificdevelopmentisnottreated completely either. Someapproaches are tentativeand shouldbe improved,some areclumsyandmightbesmoothed.Somepartsarestillunderdiscussion. After these atmospherical remarks, let us summarize the main topics that we includedandsomeofthoseexcludedfromthecontent.Wealmostentirelyfocused on the variational quantum Monte Carlo (VQMC) scheme. The diffusion Monte Carlo (DMC) topic only covers a rather trivial example, the harmonic oscillator. There is another large branch of quantum Monte Carlo calculationsfor electron systems that we entirely omit here. It is based on the path integral with explicit fermion statistics. Relying on large computing resources it is used in a model- likemannerforexampleforstrong-couplingsystemsbutrarelyappliedab-initioto systemsofmaterialscience. VQMC is usually considered as the poor man’s version of QMC primarily be- cause its theoretical concept is simple. One can refrain from the heavy complex machinery,whichishiddeninthedepthsofquantumstatistics,andcalculateon- lytheenergyexpectationvaluebyamultidimensionalintegralandminimizethe latterwithrespecttotheparameterspresentinthewavefunctionansatz.Thein- tegralitselfiscomputedwithstatisticallychosenpointsofsupport,andthatisthe Preface XI stageatwhichsomestatisticsenters.Inparticular,thereisthebeliefinthecentral limittheoremstatingthattheprocedureguaranteesthereliabilityofthosepoints whicharedrawnfromarandomwalk.Theproblemliesinanadequatechoiceofa parameterizedwavefunction.Iftherearemanyparameters,thenoneadditionally hastoutilizeregressionmethodstoobtainthebestchoiceofthem. Incontrast,thecomplexityofDMCisderivedfromtheevolutionaryschemeofa diffusionequationforthewavefunction,whichshouldconvergetowardsthetrue solution.Thus,onecan dispenseof anoptimizationprocedure.Instead,onehas toprogramthestepsoftheevolution,whichisacombinationoftheseparateac- tionsofthekineticandpotentialenergyHamiltoniansontheactualwavefunction, toobtainthesuccessiveapproximations.Thiscombinationaswellasthegenera- tionoftherandomwalkers,whichmimicthewavefunctionislesstrivial.Sowe thought it important to explain this theoretical background and to show how it workswiththoseeasygoingexamples.To satisfyoneselfwiththeroleofbeing a theoreticallypoor manwhen devotingoneself to VQMC,onecouldimaginethat DMC only replaces the optimization procedure of VQMC. One would also think thatthephysicalinsightliesinthechoiceofthefunctionalshapeofthemany-body wavefunctionratherthaninobtainingitsnumericalrepresentationasfromDMC. Actually,in scientificcalculations,one uses VQMC as the starting point and the richmanbecomesagainsuperiortothepoor. Presenting mainly VQMC in this volume, we proceed from simple examples suchasthehydrogenatom,whichhasaknownsolution,tocomplicatedonessuch asthelithiumsolid.Beinganinfinitesystem,thelatterpresentsanumberofad- ditionaltheoreticalandnumericalaspectswhichinflatethemagnitudeofthefirst example.Several intermediatesteps arethereforeinsertedandexplained:thehy- drogenmolecule,todealwithatwo-electronsystem,goingovertothreeelectrons inthelithiumatom,expandingtoanarbitrarynumberofelectronswhenenclosed inasimpleboxpotentialorwhenassembledtoanaggregateasalithiumcluster,to finallytreatingthethree-dimensionalperiodicarrayoflithiumatomsinacrystal. Thetwo-electron system providesa first glanceof particlesymmetryin thewave function.Thelithiumatomstandsformultiplicityandspinsymmetry.Insteadof localizedorbitals,planewaves areutilizedinabox,whichalso givesan opportu- nitytopresentthepair-correlationfunction.Withtheclusteroflithiumatomswe discusstheroleofaphysicalboundarythatisimportantforthecaseoftheinfinite solid because of its shape-dependent energy terms, which only slowly converge withsystemsize.Thetheoryforthesolidsuffersfromsuchtermsresultinginan unacceptableslowing-downofconvergence.Specialremedieshavetobediscussed to this end, which complicate the program structure in addition to the routines alreadyneededforasolid-statesystem. ThesolidconcludestheexamplesinthefieldofVQMCfollowedbythesubjectof DMC.Somedetailedderivationsarefoundattheendofthebookinanappendix. TheReferencescitesuggestionsfordetailsandadeeperunderstandingofthema- terialratherthanexhaustthefieldorgivehonortocontributionsfortheirhistorical importance.