ToallonthequestfortheTruth
1.3Inverselattice. ..................................................................11 1.4FirstBrillouinzone(FBZ) .....................................................14
1.7.1Originofthebandstructure..............................................21
1.7.2Born–vonKármánconditionin1D ......................................23
1.7.3 k dependenceoforbitalenergy...........................................25
1.8Atripleroleofthewavevector
1.9.1Born–vonKármánboundaryconditionin3D...........................26
1.9.2CrystalorbitalsfromBlochfunctions(LCAOCOmethod)............28
1.9.3SCFLCAOCOequations................................................31
1.9.4Bandwidth................................................................32
1.9.5Fermilevelandenergygap:insulators,metals,andsemiconductors..33
1.10Solidstatequantumchemistry.................................................39
1.10.1Whydosomebandsgoup?..............................................40
1.10.2Whydosomebandsgodown?...........................................41
1.10.3Whydosomebandsstayconstant?......................................41
1.10.4Morecomplexbehaviorexplainable–examples........................41
1.11TheHartree–Fockmethodforcrystals........................................50
1.11.1Secularequation..........................................................50
1.11.2IntegrationintheFBZ....................................................52
1.11.3Fockmatrixelements.....................................................53
1.11.4Iterativeprocedure(SCFLCAOCO)....................................55
1.11.5Totalenergy...............................................................55
1.12Long-rangeinteractionproblem...............................................56
1.12.1Fockmatrixcorrections..................................................57
1.12.2Totalenergycorrections..................................................59
1.12.3MultipoleexpansionappliedtotheFockmatrix........................61
1.12.4Multipoleexpansionappliedtothetotalenergy........................65
1.13Backtotheexchangeterm.....................................................68
1.14Choiceofunitcell..............................................................70
1.14.1Fieldcompensationmethod..............................................73
1.14.2Thesymmetryofsubsystemchoice......................................75
Chapter2:CorrelationandAnticorrelationofElectronicMotions.................81 VARIATIONALMETHODSUSINGEXPLICITLYCORRELATEDWAVE
FUNCTIONS
2.1Correlationcuspcondition .....................................................89
2.2TheHylleraasCImethod......................................................93
2.3Two-electronsystems..........................................................94
2.3.1Harmonium–theharmonicheliumatom................................94
2.3.2Highaccuracy:theJames–CoolidgeandKołos–Wolniewiczfunctions96
2.3.3Highaccuracy:neutrinomass... .........................................99
2.4ExponentiallycorrelatedGaussianfunctions .................................101
2.5Electronholes..................................................................102
2.5.1Coulombhole(“correlationhole”).......................................102
2.5.2Exchangehole(“Fermihole”)...........................................105
VARIATIONALMETHODSWITHSLATERDETERMINANTS
2.6Staticelectroncorrelation......................................................112
2.7Dynamicelectroncorrelation..................................................112
2.8Anticorrelation,ordoelectronssticktogetherinsomestates?..............118
2.9Valencebond(VB)method... .................................................126
2.9.1Resonancetheory–hydrogenmolecule.................................126
2.9.2Resonancetheory–polyatomiccase....................................129
2.10Configurationinteraction(CI)method........................................134
2.10.1Brillouintheorem.........................................................136
2.10.2ConvergenceoftheCIexpansion........................................137
2.10.3ExampleofH2 O..........................................................137
2.10.4Whichexcitationsaremostimportant?..................................140
2.10.5Naturalorbitals(NOs)–awaytoshorterexpansions..................140
2.10.6SizeinconsistencyoftheCIexpansion..................................142
2.11DirectCImethod...............................................................142
2.12MultireferenceCImethod .....................................................143
2.13Multiconfigurationalself-consistentfield(MCSCF)method ...............144
2.13.1ClassicalMCSCFapproach.............................................145
2.13.2UnitaryMCSCFmethod.................................................146
2.13.3Completeactivespace(CASSCF)methodissize-consistent..........148
NONVARIATIONALMETHODSWITHSLATERDETERMINANTS
2.14Coupledcluster(CC)method .................................................149
2.14.1Waveandclusteroperators...............................................151
2.14.2RelationshipbetweenCIandCCmethods..............................152
2.14.3SolutionoftheCCequations.............................................153
2.14.4Example:CCwithdoubleexcitations............... ....................156
2.14.5SizeconsistencyoftheCCmethod......................................158
2.15Equationofmotionmethod(EOM-CC)......................................159
2.15.1Similaritytransformation.................................................159
2.15.2DerivationoftheEOM-CCequations...................................159
2.16Many-bodyperturbationtheory(MBPT). ....................................162
2.16.1UnperturbedHamiltonian................................................162
2.16.2Perturbationtheory–slightlydifferentpresentation....................163
2.16.3MBPTmachinery–partone:energyequation..........................164
2.16.4Reducedresolventorthe“almost”inverseof
2.16.5MBPTmachinery–parttwo:wavefunctionequation..................166
2.16.6Brillouin–Wignerperturbationtheory...................................168
2.16.7Rayleigh–Schrödingerperturbationtheory..............................168
2.17Møller–PlessetversionofRayleigh–Schrödingerperturbationtheory......169
2.17.1ExpressionforMP2energy..............................................170
2.17.2IstheMP2methodsize-consistent?.....................................171
2.17.3ConvergenceoftheMøller–Plessetperturbationseries.................173
2.17.4Specialstatusofdoubleexcitations......................................174
NONVARIATIONALMETHODSUSINGEXPLICITLYCORRELATEDWAVE FUNCTIONS
2.18Møller–PlessetR12method(MP2-R12).. ....................................176
2.18.1Resolutionofidentity(RI)methodordensityfitting(DF).............177
2.18.2OtherRImethods.. .......................................................178
Chapter3:ChasingtheCorrelationDragon:DensityFunctionalTheory(DFT)...191
3.1Electronicdensity–thesuperstar.............................................194
3.2Electrondensitydistributions–Baderanalysis..... ..........................196
3.2.1Overallshapeof ρ ........................................................196
3.2.2Criticalpoints.............................................................197
3.2.3Laplacianoftheelectronicdensityasa“magnifyingglass”...........202
3.3TwoimportantHohenberg–Kohntheorems...................................204
3.3.1Correlationdragonresidesinelectrondensity:equivalenceof 0 and ρ0 .....................................................................204
3.3.2Asecretofthecorrelationdragon:theexistenceofenergy functionalminimizedby ρ0 ..............................................207
3.4TheKohn–Shamequations... .................................................211
3.4.1AKohn–Shamsystemofnoninteractingelectrons......................211
3.4.2Chasingthecorrelationdragonintoanunknownpartofthetotalenergy212
3.4.3DerivationoftheKohn–Shamequations................................213
3.5Tryingtoguesstheappearanceofthecorrelationdragon....................218
3.5.1Localdensityapproximation(LDA).....................................218
3.5.2Nonlocaldensityapproximation(NLDA)...............................219
3.5.3TheapproximatecharacteroftheDFTversusapparentrigorof abinitio computations....................................................220
3.6Onthephysicaljustificationfortheexchange-correlationenergy...........221
3.6.1Theelectronpairdistributionfunction...................................221
3.6.2Adiabaticconnection:fromwhatisknowntowardsthetarget.........222
3.6.3Exchange-correlationenergyandtheelectronpairdistribution function....................................................................226
3.6.4Thecorrelationdragonhidesintheexchange-correlationhole.........227
3.6.5Electronholesinspinresolution.........................................227
3.6.6Thedragon’sultimatehide-out:thecorrelationhole!...................229
3.6.7PhysicalgroundsfortheDFTfunctionals...............................232
3.7Visualizationofelectronpairs:electronlocalizationfunction(ELF). .......233
3.8TheDFTexcitedstates........................................................238
3.9Thehuntedcorrelationdragonbeforeoureyes..
ELECTRICPHENOMENA
4.2Themoleculeimmobilizedinanelectricfield...
4.2.1Theelectricfieldasaperturbation.......................................261
4.2.2Thehomogeneouselectricfield..........................................266
4.2.3Thenonhomogeneouselectricfield:multipolepolarizabilities andhyperpolarizabilities.. ...............................................275
4.3Howtocalculatethedipolemoment..........................................277
4.3.1Coordinatesystemdependence..........................................278
4.3.2Hartree–Fockapproximation.............................................278
4.3.3Atomicandbonddipoles. ................................................279
4.3.4Withinthezero-differentialoverlapapproximation.....................280
4.4Howtocalculatethedipolepolarizability ....................................280
4.4.1Sumoverstates(SOS)method...........................................281
4.4.2Finitefieldmethod........................................................284
4.4.3Whatisgoingonathigherelectricfields................................289
4.5Amoleculeinanoscillatingelectricfield ....................................290
MAGNETICPHENOMENA
4.6Magneticdipolemomentsofelementaryparticles................... ........294
4.6.1Electron...................................................................294
4.6.2Nucleus....................................................................295
4.6.3Dipolemomentinthefield...............................................296
4.7NMRspectra–transitionsbetweenthenuclearquantumstates.............299
4.8Hamiltonianofthesystemintheelectromagneticfield......................301
4.8.1Choiceofthevectorandscalarpotentials...............................301
4.8.2RefinementoftheHamiltonian..........................................302
4.9EffectiveNMRHamiltonian...................................................306
4.9.1Signalaveraging..........................................................307
4.9.2EmpiricalHamiltonian...................................................307
4.9.3Nuclearspinenergylevels................................................312
4.10TheRamseytheoryoftheNMRchemicalshift..............................319
4.10.1Shieldingconstants.......................................................320
4.10.2Diamagneticandparamagneticcontributions...........................321
4.11TheRamseytheoryofNMRspin–spincouplingconstants..................322
4.11.1Diamagneticcontribution.................................................322
4.11.2Paramagneticcontribution................................................323
4.11.3Couplingconstants........................................................324
4.11.4TheFermicontactcouplingmechanism.................................325
4.12Gauge-invariantatomicorbitals(GIAOs).....................................326
4.12.1Londonorbitals ...........................................................327
4.12.2Integralsareinvariant.....................................................328
THEORYOFINTERMOLECULARINTERACTIONS
5.1Ideaoftherigidinteractionenergy............................................341
5.2Ideaoftheinternalrelaxation.................................................342
5.3Interactingsubsystems.........................................................343
5.3.1Naturaldivision...........................................................343
5.3.2Whatismostnatural?.....................................................344
5.4Bindingenergy.................................................................346
5.5Dissociationenergy............................................................346
5.6Dissociationbarrier ............................................................347
5.7Supermolecularapproach......................................................347
5.7.1Accuracyshouldbethesame...
5.7.2Basissetsuperpositionerror(BSSE).. ..................................349
5.7.3Goodandbadnewsaboutthesupermolecularmethod.................350
5.8Perturbationalapproach........................................................351
5.8.1Intermoleculardistance–whatdoesitmean?...........................351
5.8.2Polarizationapproximation(twomolecules)............................352
5.8.3Intermolecularinteractions:physicalinterpretation.....................357
5.8.4Electrostaticenergyinthemultipolerepresentationplusthepenetration energy.....................................................................361
5.8.5Inductionenergyinthemultipolerepresentation.. .....................368
5.8.6Dispersionenergyinthemultipolerepresentation......................369
5.8.7Resonanceinteraction–excimers........................................376
5.9Symmetry-adaptedperturbationtheory(SAPT)..............................377
5.9.1Polarizationapproximationisillegal... ..................................377
5.9.2Constructingasymmetry-adaptedfunction.............................378
5.9.3Theperturbationisalwayslargeinpolarizationapproximation........379
5.9.4IterativeschemeofSAPT................................................381
5.9.5Symmetryforcing.........................................................385
5.9.6Alinktothevariationalmethod–theHeitler–Londoninteraction energy.....................................................................388
5.9.7Summary:themaincontributionstotheinteractionenergy............389
5.10Convergenceproblems.........................................................392
5.10.1Padéapproximantsmayimproveconvergence..........................393
5.11Nonadditivityofintermolecularinteractions.................................398
5.11.1Interactionenergyrepresentsthenonadditivityofthetotalenergy....398
5.11.2Many-bodyexpansionoftherigidinteractionenergy..................398
5.11.3Whatisadditive,whatisnot?............................................401
5.11.4Additivityoftheelectrostaticinteraction................................401
5.11.5Exchangenonadditivity... ................................................402
5.11.6Inductionnonadditivity... ................................................406
5.11.7Additivityofthesecond-orderdispersionenergy.......................409
5.11.8Nonadditivityofthethird-orderdispersioninteraction.................410
ENGINEERINGOFINTERMOLECULARINTERACTIONS
5.12Ideaofmolecularsurface......................................................411
5.12.1vanderWaalsatomicradii...............................................411
5.12.2Aconceptofmolecularsurface..........................................411
5.12.3Confiningmolecularspace–thenanovessels...........................412
5.12.4Molecularsurfaceunderhighpressure ..................................413
5.13Decisiveforces.................................................................414
5.13.1Distinguishedroleofthevalencerepulsionandelectrostaticinteraction414
5.13.2Hydrogenbond. ...........................................................415
5.13.3Coordinationinteraction..................................................417
5.13.4Electrostaticcharacterofmolecularsurface–themapsofthe molecularpotential.......................................................418
5.13.5Hydrophobiceffect .......................................................420
5.14Constructionprinciples........................................................424
5.14.1Molecularrecognition–synthons.. ......................................424
5.14.2“Key-lock,”template-like,and“hand-glove”synthoninteractions....424
5.14.3Convexandconcave–thebasicsofstrategyinthenanoscale..........427
Chapter6:ChemicalReactions...................................................437
6.1Hypersurfaceofthepotentialenergyfornuclearmotion....................442
6.1.1Potentialenergyminimaandsaddlepoints..............................443
6.1.2Distinguishedreactioncoordinate(DRC)...............................446
6.1.3Steepestdescentpath(SDP)..............................................446
6.1.4Higher-ordersaddles......................................................447
6.1.5Ourgoal...................................................................447
6.2Chemicalreactiondynamics(apioneers’approach)...
ABINITIO APPROACH
6.3Accuratesolutions(threeatoms)..............................................453
6.3.1CoordinatesystemandHamiltonian.....................................453
6.3.2SolutiontotheSchrödingerequation....................................456
6.3.3Berryphase................................................................458
APPROXIMATEMETHODS
6.4Intrinsicreactioncoordinate(IRC).. ..........................................460
6.5ReactionpathHamiltonianmethod...........................................463
6.5.1EnergyclosetoIRC......................................................463
6.5.2Vibrationaladiabaticapproximation.....................................465
6.5.3Vibrationalnonadiabaticmodel. .........................................471
6.5.4ApplicationofthereactionpathHamiltonianmethodtothereaction H2 + OH → H2 O + H...................................................473
6.6Acceptor–donor(AD)theoryofchemicalreactions........... ...............479
6.6.1Asimplemodelofnucleophilicsubstitution–MO,AD,andVB formalisms................................................................479
6.6.2MOpicture → ADpicture...............................................480
6.6.3Reactionstages............................................................484
6.6.4Contributionsofthestructuresasthereactionproceeds................489
6.6.5Nucleophilicattack–themodelismoregeneral:H + ethylene → ethylene + H ............................................................492
6.6.6Themodellooksevenmoregeneral:theelectrophilicattack H+ + H2 → H2 + H+ ...................................................495
6.6.7Themodelworksalsoforthenucleophilicattackonthepolarizedbond496
6.7Symmetry-allowedandsymmetry-forbiddenreactions......................501
6.7.1Woodward–Hoffmannsymmetryrules..................................501
6.7.2ADformalism.............................................................501
6.7.3Electrocyclicreactions....................................................502
6.7.4Cycloadditionreaction.... ................................................504
6.7.5Barriermeansacostofopeningtheclosedshells.......................508
6.8Barrierfortheelectrontransferreaction.. ....................................509
6.8.1Diabaticandadiabaticpotential..........................................509
6.8.2Marcustheory.............................................................511
6.8.3Solvent-controlledelectrontransfer......................................516
Chapter7:InformationProcessing–TheMissionofChemistry...................533
7.1Multilevelsupramolecularstructures(statics)... .............................537
7.1.1Complexsystems.........................................................537
7.1.2Self-organizingcomplexsystems........................................537
7.1.3Cooperativeinteractions.. ................................................540
7.1.4Combinatorialchemistry–molecularlibraries..........................541
7.2Chemicalfeedback–asteeringelement(dynamics)......... ................543
7.2.1Alinktomathematics–attractors.......................................543
7.2.2Bifurcationsandchaos...................................................545
7.2.3Brusselatorwithoutdiffusion... .........................................547
7.2.4Brusselatorwithdiffusion–dissipativestructures......................553
7.2.5Hypercycles...............................................................555
7.2.6Fromself-organizationandcomplexitytoinformation.................555
7.3Informationandinformedmatter..............................................556
7.3.1Abstracttheoryofinformation...........................................557
7.3.2Teachingmolecules... ....................................................559
7.3.3Dynamicinformationprocessingofchemicalwaves...................561
7.3.4Moleculesascomputerprocessors.......................................568
7.3.5Themissionofchemistry.................................................573 AppendixA:DiracNotationforIntegrals.........................................581
Whateverweknowresultsfromour interaction withtheBigThing,whichwecalltheUniverse (andwhichislargelyunknown).Whatwedoknowuntilnow,however,seemstosupportthe viewthattheUniverseitselfoperatesthroughinteractionofsomesmallobjectsknownaselementaryparticlesandthatthisinteractionitself can bedescribedinamathematicallyconsistent way.Asusual,physicsmayhaveproblemswithwhattheword“elementary”reallymeans,but forthepurposeofthisbookwemaysafelyacceptalessrigorous“elementariness,”inwhich wehavetodowithnucleiandelectronsonly.Toexplainchemistryatthequantumlevel(which isusuallymorethansatisfactoryforchemicalpractice) itissufficienttotreatallnucleiand electronsaspointchargesthatinteractelectrostatically (throughtheCoulomblaw).
“Ideasofquantumchemistry”iscomposedofvolumes1and2.Theyareaimedtobeautonomous(1onbasics,2onsomespecializedtopics)foreconomicalaswellasergonomical reasons,butatthesametimeinterrelatedandtiedbyseveralaspects(topics,structure,logical connections,appendices)thusforminganentity.
“Ideasofquantumchemistry”Volume1:Fromquantumphysicstochemistry focusesonmaking twoimportantapproximations withthegoalofbeingabletoexplaintheverybasicsof chemistry.Themostimportantistheconceptofthemolecularthree-dimensionalstructure(resultingfromtheBorn–Oppenheimerapproximation,ChapterV1-6).Asecondapproximation madeassumesthateachelectronisdescribedbyitsownwavefunction(“electronicorbital,” ChapterV1-8).Toarriveattheseapproximationswithinthenonrelativisticapproachwefirst exploitthefactthateventhelightestnucleus(proton)ismorethanathousandtimesheavier thananelectron.AsaconsequencetheBorn–Oppenheimerapproximationallowsonetosolve theproblemofelectronicmotionassumingthatthenucleiaresoheavythattheydonotmove, i.e.,theyoccupycertainpositionsinspace.Theorbitalapproximationmeansthatanyelectron seesotherelectrons’motionaveraged.Fromthis,however,followsthatamovingelectrondoes notreacttosomeparticularpositionsofotherelectrons,whichlooksasiftheydidnotseeeach other’spositions.
Bymakingapproximationsofthiskind,wehavearrivedata(“minimal”)modelofthemolecule asanentitythathasadistinctspatialshape(“geometry”)andiskepttogetherbecauseofthe
Introduction(2)
chemicalbondsbetweenpairsofneighboringatoms.Usingtheminimalmodelwewereable toexplainwhychemicalbondscanbeformed.Suchasystemofbondsisstablewithrespect torelativelysmalldeviationsofftheequilibriummoleculargeometry(“chemicalbondpattern staysunchanged”)definedasthosepositionsofthenucleiwhichensurethelowestelectronic ground-stateenergyofthesystem(potentialenergysurface[PES]forthemotionofthenuclei). Largerdeviationsmovethesystemtowardsotherchemicalbondpatterns(stillwithinthesame electronicgroundstate1 ),whichmayalsohavetheirstableBorn–Oppenheimerequilibrium structures.Theatomsvibrateabouttheirequilibriumpositions,whilethemoleculeasawhole fliesinspacealongastraightlineandkeepsrotatingaboutitscenterofmass.
Thus,anymoleculecanbetreatedtoafirstapproximationasanobject,small,butsimilar totheobjectsweencounterineverydaylife.Ithasastablealbeitflexiblethree-dimensional architecture,wemayspeakaboutitsdetailslikeleft-andright-handside,itsfront,itsback, etc. Thispicturerepresentstheveryfoundationofallbranchesofchemistryandbiologyas wellasalargeportionofphysics. Understandingchemistrywouldbeextremelydifficult,if notimpossible,withoutthiscrucialpicture.Wehavetoremember,however,thattherealityis morecomplicatedthanthat,andthisisasimplificationonly,anextremelyfertileandfortuitous approximation.
Theminimalmodelworksusuallywithanerrorofabout1%intotalenergyandequilibrium atom–atomdistances.Thisseemsatfirstsightassatisfactory,anditisforabigportionof chemistry.However,themodelfailsspectacularlyforseveralimportantphenomena,likedescriptionofmetals,dissociationofachemicalbond,ubiquitousandimportantintermolecular interactions,etc.
“Ideasofquantumchemistry”Volume2:Interactions improvestheminimalmodelofa moleculebytakingintoaccounttheelectron–electronmutualcorrelationofmotion(Chapters 2 and 3)aswellasthemolecule’sinteractionwiththeexternalworld:inacrystal(Chapter 1),in externalelectricandmagneticfields(Chapter 4),andwithothermolecules(Chapters 5 and 6).
Theauthorisconvincedthatchemistryfacescurrentlythechallengeofinformationprocessing, quitedifferenttothisperformedbyourcomputers.Thisperspective,addressedmainlytomy youngstudents,isdiscussedinthelastchapter(Chapter 7)ofthisbook,whichdiffersvery muchfromotherchapters.Itshowssomeexcitingpossibilitiesofchemistryandoftheoretical chemistry,andposessomegeneralquestionsastowherethelimitsaretobeimposedondevelopmentscienceinordertobehonestwithrespecttoourselves,tootherpeople,andtoour planet.
Theideaoforbitalsisnotonlyasimpleandfruitfuldescriptionofmoleculeswithintheminimalmodel.Mostimportantly,theorbitalmodelprovidesalsoamapofideasandavocabulary,
1 HigherinenergyscaleareexcitedPESs.
Introduction(2) whichareusedbeyondtheorbitalmodel,inanymoresophisticatedtheory.Strictlyspeaking, inanyadvancedtheoryintroducedinvolume2,thereisnosuchthingasadefinitionofelectronicorbital,butstillthe“orbitallanguage”ismostoftenusedinthismorecomplexsituation, becauseitissimple,flexible,andinformative.
Volume1maybeviewedasindependentofvolume2,i.e.,volume1isautonomous,ifthe minimalmodelofchemistryisthereader’starget.Theoppositeisnottrue:volume2relieson volume1,mainlybecauseitneedstheminimalmodelasthestartingpoint.Tokeepvolume2as autonomicaspossibletheorbitalmodel(theHartree–Focktheory)isbrieflyrepeated(mainly Appendices A and B).
TREE
Anybookhasalinearappearanceandthepagenumbersremindusofthat.However,the logic ofvirtuallyanybookis nonlinear,andinmanycasescanbevisualizedbyadiagramconnectingthechaptersthat(logically)followfromoneanother.Suchadiagramallowsformultiple branchesemanatingfromagivenchapter,ifbeingonequalfooting.Thelogicalconnectionsare illustratedinthisbookasaTREEdiagram,playinganimportantroleinourbookandintended tobeastudyguide.Anauthorleadsthereaderinacertaindirectionandthereaderexpects tobeinformedwhatthisdirectionis,whythisdirectionisneeded,whatwillfollowthen,and whatbenefitshe/shewillgainaftersuchastudy.
AthicklineinthecenteroftheTREEseparatesvolume1(bottompart)fromvolume2(upper part).
Thetrunkrepresentsthepresentbook’sbackboneandiscoveredbythecontentofvolume1.
• Itbeginsbypresentingthefoundationofquantummechanics(postulates).
• ItcontinueswiththeSchrödingerequationforstationarystates,sofarthemostimportant equationinquantumchemicalapplications,and
• theseparationofnuclearandelectronicmotion(throughtheadiabaticapproximation, the centralideaofthepresentbookandchemistryingeneral).
• Itthendevelopstheorbitalmodelofelectronicstructure.
Thetrunkthuscorrespondstoatraditionalcourseinquantumchemistryforundergraduates. ThismaterialrepresentsthenecessarybasisforfurtherextensionsintootherpartsoftheTREE (appropriateforgraduatestudents).ThetrunkmakesitpossibletoreachthecrownoftheTREE (volume2),wherethereadermayfindtastyfruit.
