高等固体物理-关联PPT课件

.,1,第五章关联,5.1单电子近似的理论基础5.2费米液体理论5.3强关联体系,.,2,多电子体系(AfterBorn-Oppenheimer绝热近似):,5.1单电子近似的理论基础,关联：电子电子相互作用,弱：单电子近似，电子平均场,.,3,1.Hartree方程(1928),连乘积形式：,按变分原理，的选取E达到极小,正交归一条件,单电子方程,.,4,动能,原子核对电子形成的势能,其余N-1个电子对j电子的库仑作用能,自洽求解，H2,He计算与实验相符。26个电子的Fe原子，运算要涉及1076个数，对称简化1053个整个太阳系没有足够物质打印这个数据表！,2.凝胶模型(jelliummodel),为突出探讨相互作用电子系统的哪些特征是区别于不计其相互作用者，可人为地简化假定电子是沉浸在空间密度持恒的正电荷背景之中(不考虑离子的周期性)。正电荷的作用体现于在相互作用电子体系的Hamiltonian中出现一个维持系统聚集的附加项,金属体系，设电子波函数：,.,5,Hartree方程中的势：,第二项是全部电子在r处形成的势，与相抵消第三项是须扣除的自作用，与j有关，但如取r为计算原点：,所以对凝胶模型，Hartree方程：,相互作用没有相互作用电子正电荷背景自由电子气,.,6,3.Hartree-Fock方程(1930)Hartree方程不满足Pauli不相容原理电子：费米子单电子波函数f：N电子体系的总波函数：,不涉及自旋轨道耦合时：,N电子体系能量期待值：,1.第二项j,j可以相等，自相互作用2.自相互作用严格相消（通过第二，三项）3.第三项为交换项，同自旋电子,.,7,通过变分:,么正变换:,单电子方程：,与Hartree方程的差别：第三项对全体电子，第四项新增，交换作用项。求和只涉及与j态自旋平行的j态，是电子服从Fermi统计的反映。,4.Koopmann定理（1934),单电子轨道能量等于N电子体系从第j个轨道上取走一个电子并保持N1个电子状态不不变的总能变化值。,.,8,推广：系统中一个电子由状态j转移到态i而引起系统能量的变化,5.交换空穴(Fermihole)将H-F方程改写为：,其中：,.,9,定性讨论：假设,Fermihole:与某电子自旋相同的其余邻近电子在围绕该电子形成总量为1的密度亏欠域,.,10,举例：利用HF方程和Koopmann定理研究原子的电离能,.,11,Nonlocalexchangeterm,HF方法的缺点：HF方程仅考虑了交换作用，没有考虑关联，许多体系不能正确描述（金属）对原子/分子体系计算量不算大，但对固体计算量就很大：,用平均的局域（交换）势替代HF方程中的非局域交换势：,Xa,6.SlatersXa方法,.,12,energyasafunctionoftheoneelectrondensity,nuclear-electronattraction,electron-electronrepulsionThomas-FermiapproximationforthekineticenergySlaterapproximationfortheexchangeenergy,7.密度泛函理论(Densityfunctionaltheory),(1)Thomas-Fermi-DiracModel,.,13,Thomas-FermiModel,Thomas-Fermimodel(semiclassical):1927Electrondensityofauniformelectrongas:Fermiwavevectorandelectronkineticenergyofauniformelectrongas,.,14,Wemayassumethatthekineticenergyoftheelectrongasdependsonthelocalelectrondensity:Thetotalkineticenergyofelectronsinthesystemisthereforeafunctionalofelectrondensity:,Thomas-FermiModel,.,15,(2)TheHohenberg-KohnTheorem,propertiesareuniquelydeterminedbytheground-stateelectron,In1964,HohenbergandKohnprovedthat,molecularenergy,wavefunctionandallothermolecularelectronic,probabilitydensity,namely,Phys.Rev.136,13864(1964),.”,Densityfunctionaltheory(DFT)attemptsto,andotherground-statemolecularproperties,fromtheground-stateelectrondensity,“Formoleculeswithanondegenerategroundstate,theground-state,calculate,.,16,Proof:,TheelectronicHamiltonianis,itisproducedbychargesexternaltothesystemofelectrons.,InDFT,iscalledtheexternalpotentialactingonelectroni,since,Oncetheexternalpotential,theelectronicwavefunctionsandallowedenergiesofthemoleculeare,andthenumberofelectronsnarespecified,determinedasthesolutionsoftheelectronicSchrdingerequation.,.,17,Nowweneedtoprovethattheground-stateelectronprobabilitydensity,thenumberofelectrons.,theexternalpotential(exceptforanarbitraryadditiveconstant),a)Since,determinesthenumberofelectrons.,b)Toseethat,determinestheexternalpotential,wesuppose,thatthisisfalseandthattherearetwoexternalpotentials,and,(differing,bymorethanaconstant)thateachgiverisetothesameground-stateelectron,density,.,determines,.,18,theexactground-statewavefunctionandenergyof,theexactground-statewavefunctionandenergyof,Let,Since,and,differbymorethanaconstant,and,mustbedifferentfunctions.,.,19,Proof:,Assume,thus,thus,whichcontradictsthegiven,information.,function,theexactground-statewavefunction,stateenergy,foragivenHamiltonian,Ifthegroundstateisnondegenerate,thenthereisonlyonenormalized,thatgivestheexactground,.,20,Accordingtothevariationtheorem,supposethat,If,then,isanynormalized,well-behavedtrialvariationfunction.,Nowuse,asatrialfunctionwiththeHamiltonian,then,Substituting,gives,.,21,Let,beafunctionofthespatialcoordinates,ofelectroni,then,Usingtheaboveresult,weget,Similarly,ifwegothroughthesamereasoningwithaandbinterchanged,weget,.,22,Byhypothesis,thetwodifferentwavefunctionsgivethesameelectron,.Putting,andaddingtheabovetwoinequalities,density:,yield,potentialscouldproducethesameground-stateelectrondensitymustbefalse.,energy)andalsodeterminesthenumberofelectrons.,Thisresultisfalse,soourinitialassumptionthattwodifferentexternal,potential(towithinanadditiveconstantthatsimplyaffectsthezerolevelof,Hence,theground-stateelectronprobabilitydensity,determinestheexternal,.,23,probabilitydensity,andotherproperties”,emphasizesthedependenceof,theexternalpotential,differsfordifferentmolecules.,“Forsystemswithanondegenerategroundstate,theground-stateelectron,determinestheground-statewavefunctionandenergy,which,However,thefunctionals,areunknown.,isalsowrittenas,Thefunctional,independentoftheexternal,on,is,potential.,.,24,(3)TheHohenberg-kohnvariationaltheorem,“Foreverytrialdensityfunction,thatsatisfies,and,forall,thefollowinginequalityholds:,isthetruegroundstateenergy.”,Proof:,Let,satisfythat,and,Hohenberg-Kohntheorem,determinestheexternalpotential,andthisinturndeterminesthewavefunction,density,.Bythe,thatcorrespondstothe,.,where,.,25,withHamiltonian,.Accordingtothevariationtheorem,Letususethewavefunction,asatrialvariationfunctionforthemolecule,Sincethelefthandsideofthisinequalitycanberewrittenas,Onegets,states.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.,HohenbergandKohnprovedtheirtheoremsonlyfornondegenerateground,.,26,(4)TheKohn-Shammethod,Ifweknowtheground-stateelectrondensity,molecularpropertiesfrom,function.,theHohenberg-Kohn,theoremtellsusthatitispossibleinprincipletocalculatealltheground-state,withouthavingtofindthemolecularwave,1965,KohnandShamdevisedapracticalmethodforfinding,and,forfinding,from,.Phys.Rev.,140,A1133(1965).Theirmethod,iscapable,inprinciple,ofyieldingexactresults,butbecausetheequationsof,theKohn-Sham(KS)methodcontainanunknownfunctionalthatmustbe,approximated,theKSformationofDFTyieldapproximateresults.,沈吕九,.,27,electronsthateachexperiencethesameexternalpotential,theground-stateelectronprobabilitydensity,equaltotheexact,ofthemoleculeweareinterestedin:,.,KohnandShamconsideredafictitiousreferencesystemsofnnoninteracting,thatmakes,ofthereferencesystem,Sincetheelectronsdonotinteractwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemis,where,istheone-electronKohn-ShamHamiltonian.,.,28,Thus,theground-statewavefunction,ofthereferencesystemis:,isaspinfunction,orbitalenergies.,areKohn-Sham,Forconvenience,thezerosubscripton,isomittedhereafter.,Define,asfollows:,ground-stateelectronickineticenergy,systemofnoninteractingelectrons.,(either,),isthedifferenceintheaverage,betweenthemoleculeandthereference,Thequantity,repulsionenergy.,units)fortheelectrostaticinterelectronic,istheclassicalexpression(inatomic,.,29,Rememberthat,Withtheabovedefinitions,canbewrittenas,Definetheexchange-correlationenergyfunctional,by,Nowwehave,sideareeasytoevaluatefrom,getagoodapproximationto,totheground-stateenergy.,Thefourthquantity,accurately.ThekeytoaccurateKSDFT,calculationofmolecularpropertiesisto,Thefirstthreetermsontheright,isarelatively,smallterm,butisnoteasytoevaluate,andtheymakethemaincontributions,.,30,Thus,becomes,.,Nowweneedexplicitequationstofindtheground-stateelectrondensity,.,sameelectrondensityasthatinthegroundstateofthemolecule:,isreadilyprovedthat,Sincethefictitioussystemofnoninteractingelectronsisdefinedtohavethe,it,.,31,ground-stateenergybyvarying,tominimizethefunctional,canvarytheKSorbitals,minimizetheaboveenergyexpressionsubjecttotheorthonormalityconstraint:,TheHohenberg-Kohnvariationaltheoremtellusthatwecanfindthe,soas,.Equivalently,insteadofvarying,we,Thus,theKohn-Shamorbitalsarethosethat,withtheexchange-correlationpotential,definedby,(If,isknown,itsfunctionalderivative,isalsoknown.),.,32,TheKSoperator,exchangeoperatorsintheHFoperatorarereplacedby,theeffectsofbothexchangeandelectroncorrelation.,isthesameastheHFoperatorexceptthatthe,whichhandles,(5)Localdensityapproximation,Comparetouniformelectrongasresult:,.,33,LetscomparethePZparameterizationwiththeuniformelectrongasresultforrsHessian矩阵，2n+1定理冻声方法，分子动力学谱分析方法,.,66,几何Berry位相,电介质极化，介电常数偶极矩-宏观极化；流-极化变化电荷密度（波函数的模）；流（波函数的位相）零电场情况下，任意两个晶体态之间的极化变化对应着一个几何量子位相晶格振动、铁电、压电效应、自发极化、静态介电张量、电子介电常数。不如传统的微扰理论方法普适，但实现简单、计算量小,.,67,PartII:数值方法,.,68,数值离散方法,基组展开LCAO基组（Gaussian基组、数值基组）实空间网格,.,69,平面波基组：从OPW到PP,平面波展开正交化平面波（OPW）赝势（PP）方法经验赝势模守恒赝势超软赝势,.,70,Muffin-tin势场与分波方法,Muffin-tin势场近似缀加平面波（APW）格林函数方法（KKR）线性化方法LAPWLMTO分波方法的发展FP-LAPWthird-generationMTO,NMTO,EMTO,.,71,平面波基组：从USPP到PAW,投影缀加波（PAW）方法赝波函数空间USPPorPAW?(VASP,ABINIT,.),.,72,实空间网格,简单直观允许通过增加网格密度系统地控制计算收敛精度线性标度可以方便的通过实空间域分解实现并行计算处理某些特殊体系（带电体系、隧穿结。）,.,73,有限差分,从微分到差分提高FD方法的计算效率对网格进行优化，如曲线网格（适应网格）和局部网格优化（复合网格）结合赝势方法多尺度（multiscale）或预处理（preconditioning）,.,74,有限元,变分方法处理复杂的边界条件矩阵稀疏程度及带状结构往往不如有限差分好广义的本征值问题,.,75,多分辨网格上的小波基组,多分辨分析半取样（semicardinal）基组,.,76,线性标度与量子力学中的局域性,“近视原理”局域化的Wannier函数或密度矩阵绝缘体：指数衰减，能隙越大衰减越快金属：零温下按幂率衰减，在有限温度下可出现指数衰减局域区域线性标度系数，crossover,.,77,线性标度算法,分治方法费米算符展开和费米算符投影方法直接最小化方法密度矩阵最小化轨道最小化优基组密度矩阵最小化,.,78,线性标度算法,基于格林函数的递归方法脱离轨道的（orbital-free,OF）算法对角化以外的线性标度构造有效哈密顿量的算法几何优化与分子动力学TDDFT,.,79,PartIII:应用,.,80,物理学：强相关体系,模型哈密顿量LDA+电子结构：CrO2点阵动力学:钚,.,81,化学：弱作用体系,松散堆积的软物质、惰性气体、生物分子和聚合物，物理吸附、Cl+HD反应用传统的密度泛函理论处理弱作用体系一个既能产生vdW相互作用系数又能产生总关联能的非局域泛函：无缝的（seamless）方法GW近似密度泛函加衰减色散（DFdD）,.,82,生命科学：生物体系,困难（尺寸问题、时间尺度）QM/MM方法（饱和原子法、冻结轨道法）简单势能面方法线性同步过渡（LST）二次同步过渡（QST）完全的分子动力学并行复制动力学（parallelreplicadynamics）超动力学（hyperdynamics,metadynamics）温度加速的动力学（temperatureaccelerateddynamics）快速蒙特卡罗（on-the-flykinericMonteCarlo）方法,.,83,纳米和材料科学：输运性质及其他,输运：非平衡态第一性原理模拟材料力学：运动学MonteCarlo（KMC）-点阵气体和元胞自动机-连续方程的有限差分有限元求解,.,84,光谱学：激发态和外场,系综密度泛函理论考虑系统对称性，用求和方法计算多重态激发能多体微扰理论，GW近似Bethe-Salpeter方程TDDFT，线性响应,.,85,一些计算软件,Gaussian,DMol3,Q-Chem,ADF,SIESTAVASP,CASTEP,ABINIT,PWSCF,CPMDOctopusBigDFT,.,86,5.2费米液体理论,费米体系费米温度：,均匀的无相互作用的三维系统，费米温度：,费米简并系统：费米子系统的温度通常运运低于费米温度室温下金属中的传导电子,费米温度给出了系统中元激发存在与否的标度在费米温度以下，系统的性质由数目有限的低激发态决定。有相互作用和无相互作用的简并费米子系统中，低激发态的性质具有较强的对应性。,.,87,2.费米液体金属中电子通常是可迁移的，称为电子气，电子动能：,电子势能：,在高密度下，电子动能为主，自由电子气模型是较好的近似。在低密度下，电子之间的势能或关联变得越来越重要，电子可能由于这种关联作用进入液相甚至晶相。较强关联下，电子系统被称为电子液体或费米液体或Luttinger液体(1D),.,88,相互作用：(1)单电子能级分布变化(势的变化);(2)电子散射导致某一态上有限寿命(驰豫时间),3.朗道费米液体理论单电子图象不是一个正确的出发点，但只要把电子改成准粒子或准电子，就能描述费米液体。准粒子遵从费米统计，准粒子数守恒，因而费米面包含的体积不发生变化。,假设激发态用动量表示,.,89,朗道费米液体理论的适用条件：(1).必须有可明确定义的费米面存在,(2).准粒子有足够长的寿命,.,90,FermiLiquidTheory,whereisthenumberoperatorinmomentumspace,Firstofall,WhatisFermiLiquid?IfasystemisdescribedbythefollowingHamiltonian,itiscalledFermiLiquid:,(1)Thetheoryisnon-trivialbecauseofthequarticinteraction(2)However,thesingle-particlepicturemightworkundersomeassumptions,.,91,Considerasystemwith4fermioninteraction.ThemostgeneralHamiltonianis:,Inweakcoupling,someinteractionsaremoreimportantthantheotherbyspaceargument.Take2Delectrongasasanexample.Intheweakcouplingandlowtemperaturelimit,onlyelectronsveryclosetotheFermisurfaceisimportant.,.,92,Duetomomentumconservation,notallanglesarefree,Takeasindependentparameterfirst.Forgiven,theothertwomomentaarecompletelyfixed.However,if,theallowedphasespaceisenlargedtothewholeFermisurface.Thus,weexpectthiskindofinteractionwoulddominateovertheothers.,BCSscattering,Notethatthisargumentisonlyappropriateinweakcouplinglimit!,.,93,Ontheotherhand,ifwetakeasindependentparameter,thereisanotherkindofinteractionwoulddominatetheothers(bysimilarargument),forwardscattering,Thus,inweakcoupling,weexpectthesetwointeractionsarethemostimportantones.,(1)Forrepulsiveinteraction,itcanbeshownthattheBCSinteractioncanbesafelyignoredFermiLiquid!(2)Forattractiveinteraction,theinstabilityistriggeredThesystemisbetterdescribedbyBCStheory!,.,94,NowwearereadytowritedowntheFermiLiquidTheoryinmorefamiliarform,Inweakcouplingandatlowtemperature,thedensityisnotfarfromthefreeFermidistribution.Thus,definethedensityvariation,RewriteHintermsof,.,95,Bareenergy:,Itisonlyaconstant.Willignoreitlater.,quasi-particlespectrum:,Finally,quasi-particleinteractions,Collectingallterms,theHamiltonianis,.,96,Nowusemean-fieldapproximation,wecansolvethequasti-particledistributionself-consistently.,Itbecomesaquadratictheoryandthemean-fielddispersionis:,MakeuseoftheMFdispersion,weobtaintheself-consistentequation:,.,97,SimplePictureforFermiLiquid,Freetheory:,Now,addintheparticle-particleinteraction.,Onequasi-particle:,Manyquasi-particlespresent:,.,98,Manyinterestingproperties:,(1)Effectivemass,(2)Specificheat:,(3)Soundvelocity:,(4)Spinsusceptibility:,Lineartemperaturedependence,ComparewithCuriesusceptibility,.,99,朗道费米液体理论是处理相互作用费米子体系的唯象理论。在相互作用不是很强时，理论对三维液体正确。二维情况下，多大程度上成立不知道。一维情况下，不成立。luttinger液体一维：低能激发为自旋为1/2的电中性自旋子和无自旋荷电为的波色子的激发。,非费米液体行为：与费米液体理论预言相偏离的性质,.,100,THEPHYSICSOFLUTTINGERLIQUIDS,FERMISURFACEHASONLYTWOPOINTS,failureofLandausFermiliquidpicture,ELECTRONSFORMAHARMONICCHAINATLOWENERGIES,Coulomb+Pauliinteraction,THELUTTINGERLIQUID:INTERACTINGSYSTEMOF1DELECTRONSATLOWENERGIES,collectiveexcitationsarevibrationalmodes,.,101,REMARKABLEPROPERTIES,.,102,OUTLINE,WhatisaFermiliquid,andwhytheFermiliquidconceptbreaksin1D,.,103,LITERATURE,H.J.Schulz,G.CunibertiandP.PieriFermiliquidsandLuttingerliquids,cond-mat/9807366,.,104,SHORTLYABOUTFERMILIQUIDS,Landau1957-1959,Lowenergyexcitationsofasystemofinteractingparticlesdescribedintermsofquasi-particles(single-particleexcitations),.,105,FERMILIQUIDSII,PauliexclusionprincipleonlystateswithinkTaroundFermisphereavailablequasiparticlestatesnearFermispherescatteronlyweakly,QUASI-PARTICLEPICTUREISAPPLICABLEIN3D,EffectofCoulombinteractionistoinduceafinitelife-timet,3D,.,106,FERMILIQUIDSIII,.,107,LIFETIMEOFQUASI-PARTICLES,scatteringoutofstatek,scatteringintostatek,spin,screenedCoulombinteraction,energyconservation,In3Danintegrationoverangulardependencetakescareofd-function,Fermisgoldenruleyieldsforthelifetimet,T=0,.,108,LIFETIMEOFQUASI-PARTICLESII,In1Dk,karescalars.Integrationoverkyields,Whataboutthelifetimetin1D?,AtsmallT,i.e.,thisratiocannotbemadearbitrarilysmallasin3D,.,109,BREAKDOWNOFLANDAUTHEORYIN1D,1,2,3,4,0,1,3,2,DISPERSIONOFEXCITATIONSIN1D,(RPA),singleparticle,gaplessplasmon,COLLECTIVEANDSINGLE-PARTICLEEXCITATIONNONDISTINCT,nolongerdivergesat(noangularintegrationoverdirectionofasin3D),.,110,THETOMONAGA-LUTTINGERMODEL,EXACTLYSOLVABLEMODELFORINTERACTING1DELECTRONSATLOWENERGIES,.,111,TOMONAGA-LUTTINGERHAMILTONIAN,.,112,TLHAMILTONIANII,Interactions,freepart,interaction,.,113,BOSONIZATION,BOSONIZATION:EXPRESSFERMIONICHAMILTONIANINTERMSOFBOSONICOPERATORS,constructbosonicHamiltonianwiththesamespectrun,(a)and(b)havesamespectrumbutdifferentgroundstate,EXCITEDSTATECANBEWRITTENINTERMSOFCHARGEEXCITATIONS,ORBOSONICELECTRON-HOLEEXCITATIONS,.,114,STEP1,WHICHOPERATORSDOTHEJOB?,Introducethedensityoperators(createexcitationofmomentumq),andconsidertheircommutationrelations,nearlybosoniccommutationrelations,.,115,STEP1:PROOF,Considere.g.,algebraoffermionicoperators,occupationoperator,.,116,STEP2,Examinenow,BOSONIZEDHAMILTONIAN,STATESCREATEDBYAREEIGENSTATESOFWITHENERGY,and,interactions,.,117,STEP2:PROOF,Example:,.,118,STEP3,Introducethebosonicoperators,yielding,DIAGONALIZATION,.,119,SPIN-CHARGESEPARATION,andinteraction(satisfyingSU2symmetry),Ifweincludespin,itgetsslightlymorecomplicated.andinteresting,Introducethespinandchargedensities,Hamiltoniandecoupleintwoindependentspinandchargeparts,withexcitationspropagatingwithvelocities,.,120,SPACEREPRESENTATION,Longwavelengthlimit(interactions),AppropriatelinearcombinationsP,qofthefieldr(x)canbedefined.Thenonefinds,where,Luttingerparameterg5f3d4d5d_能带宽度上升另外，从左往右穿过周期表，部分填充壳层的半径逐步降低，关联重要性增加。,.,129,4f，5f元素和3d,4d,5d元素的壳层体积与Winger-Seitz元胞体积的比值,Y,Sc,.,130,Smith和Kmetko准周期表,窄带区域,重费米子强铁磁性超导体,离域性,局域性,.,131,另一类窄带现象：来自能带中的近自由电子与溶在晶格中具有3d,5f或4f壳层电子的溶质原子相互作用Friedel与Anderson稀土元素或过渡金属化合物中的能隙不可能仅用“电荷转移能”、“杂化能隙”、“有效库仑相关能”三者之一来描述，而应该说三者同时发挥作用。稀土化合物部分存在混价“mixedvalence”。混价的作用导致在Fermi面附近存在非常窄的能带(部分填充f能带或f能级），电子可以在4f能级和离域化能带之间转移，对固体基态性质产生显著影响。,.,132,2.窄能带现象的理论模型,选择经验参数的模型Hamilton量方法Hubbard模型和Anderson模型,.,133,TheHubbardModel,Fromsimplequantummechanicstomany-particleinteractioninsolids-ashortintroduction,.,134,Historicalfacts,HubbardModelwasfirstintroducedbyJohnHubbardin1963.WhowasHubbard?Hewasbornin1931anddied1980.Theoreticianinsolidstatephysics,fieldofwork:Electroncorrelationinelectrongasandsmallbandsystems.HeworkedattheA.E.R.E.,Harwell,U.K.,andattheIBMResearchLabs,SanJos,USA.,Picturetakenfrom:PhysicsToday,Vol.34,No4,1981,.,135,What,ingeneral,istheHM?,Hubbardmodelisaquantumtheoreticalmodelformany-particleinteractioninandwithaperiodiclatticeItisbasedonaninteractionHamitonian,sometransformationsandassumptionstobeabletotreatcertainproblems(e.g.magneticbehaviourandphasetransitions)withsolidstatetheory,.,136,Quantummechanics,Basics:SchrdingerequationExpectationvaluesOrthonormalityandclosurerelationThebra-ketnotation,.,137,Basistransformation,mathematically,Abasistransformationcanbesimplyperformed:,Anequationistransformedthesameway:,.,138,Singleparticleequations,Particleinapotential:Periodicpotentials:Solutionforweakcouplingtopotential:Blochwave,.,139,Singleparticleequations,Dispersionrelationforfreeelectrons(dashedline):,DispersionrelationforBlochelectrons(quasi-free)(solidline):Theenergiesatarenolongerdegenerated.Twoeigenenergiesatthosepoints.,GraphfromGerdCzycholl,TheoretischeFestkrperphysik“,Vieweg-Verlag,.,140,Singleparticleequations,Wannierstatesproduceanorthonormalbaseoflocalizedstates;atomicwavefunctionswouldalsobelocalized,buttheyarenotorthonormal.,Strongerlatticepotential:couplingtolatticepointsoccurs;amodifiedBlochwaveisused,e.g.WannierstatesresultingfromtheTight-Binding-Model:,.,141,Comparisonbetweenthetwonewwavefunctions,Blochwavefunction,Wannierwavefunction(w-part),GraphfromGerdCzycholl,TheoretischeFestkrperphysik“,Vieweg-Verlag,GraphfromGerdCzycholl,TheoretischeFestkrperphysik“,Vieweg-Verlag,.,142,Wavefunctionformanyparticles,Wavefunctionisnotsimplytheproductofallsingleparticlewavefunctions;Particlescannotbedi