香港大学量子化学.pptx

ComputationalChemistryG.H.CHENDepartmentofChemistryUniversityofHongKongIn1929,Diracdeclared,“Theunderlyingphysicallawsnecessaryforthemathematicaltheoryof.thewholeofchemistryarethuscompletelyknow,andthedifficultyisonlythattheexactapplicationoftheselawsleadstoequationsmuchtoocomplicatedtobesoluble.”BeginningofComputationalChemistryTheoryisreality!W.A.GoddardIIIDiracComputationalChemistryQuantumChemistryMolecularMechanicsBioinformaticsCreate&AnalyseBio-informationSchrdingerEquationF=MaMulliken,1966Fukui,1981Hoffmann,1981Pople,1998Kohn,1998NobelPrizesforComputationalChemsitryComputationalChemistryIndustryCompanySoftwareGaussianInc.Gaussian94,Gaussian98SchrdingerInc.JaguarWavefunctionSpartanQ-ChemQ-ChemAccelrysInsightII,Cerius2HyperCubeHyperChemCeleraGenomics(Dr.CraigVenter,formalProf.,SUNY,Baffalo;98-01)Applications:materialdiscovery,drugdesign&researchR&DinChemical&Pharmaceuticalindustriesin2000:US$80billionBioinformatics:TotalSalesin2001US$225millionProjectSalesin2006US$1.7billionLODESTARv1.02-LocalizedDensityMatrix:STARperformerhttp:/yangtze.hku.hkSoftwareDevelopmentatHKUQuantumChemistryMethodsAbinitiomolecularorbitalmethodsSemiempiricalmolecularorbitalmethodsDensityfunctionalmethodH=E SchrdingerEquationHamiltonianH=(-h2/2m)2 2-(-(h2/2me)i i2+Z Z e2 2/r-i Z Z e2/ri +i j e2/rijWavefunctionEnergyVitaminCC60CytochromechemeOH+D2-HOD+DenergyC60andSuperconductorApplications:Magnet,Magnetictrain,PowertransportationWhatissuperconductor?ElectricalCurrentflowsforever!CrystalStructureofC60solidCrystalStructureofK3C60K3C60isaSuperconductor(Tc=19K)Erwin&Pickett,Science,1991GHChen,Ph.D.Thesis,Caltech(1992)VibrationSpectrumofK3C60EffectiveAttraction!ThemechanismofsuperconductivityinK3C60wasdiscoveredusingcom-putationalchemistrymethodsVarmaet.al.,1991;Schluteret.al.,1992;Dresselhauset.al.,1992;Chen&Goddard,1992CarbonNanotubes(Ijima,1991)STMImageofCarbonNanotubes(Wildoeret.al.,1998)CalculatedSTMImageofaCarbonNanotube(Rubio,1999)ComputerSimulations(Saito,Dresselhaus,Louieet.al.,1992)CarbonNanotubes(n,m):Conductor,ifn-m=3II=0,1,2,3,;orSemiconductor,ifn-m 3IMetallicCarbonNanotubes:ConductingWiresSemiconductingNanotubes:TransistorsMolecular-scalecircuits!1nmtransistor!0.13mtransistor!30nmtransistor!Wildoer,Venema,Rinzler,Smalley,Dekker,Nature391,59(1998)ExperimentalConfirmations:Lieberet.al.1993;Dravidet.al.,1993;Iijimaet.al.1993;Smalleyet.al.1998;Haddonet.al.1998;Liuet.al.1999Science9thNovember,2001Logicgates(andcircuits)withcarbonnanotucetransistorScience7thJuly,2000Carbonnanotube-BasednonvolatileRAMformolecularcomputingNanoelectromechanicalSystems(NEMS)K.E.Drexler,Nanosystems:MolecularMachinery,ManufacturingandComputation(Wiley,NewYork,1992).LargeGearDrivesSmallGearG.Honget.al.,1999Nano-oscillatorsZhao,Ma,Chen&Jiang,Phys.Rev.Lett.2003NanoscopicElectromechanicalDevice(NEMS)HibernationAwakeningOscillationQuantummechanicalinvestigationofthefieldemissionfromQuantummechanicalinvestigationofthefieldemissionfromthetipsofcarbonnanotubesthetipsofcarbonnanotubesZettl,PRL2001Zheng,Chen,Li,Deng&Xu,Phys.Rev.Lett.2004Computer-AidedDrugDesignGENOMICSHumanGenomeProjectALDOSEREDUCTASEDiabetesDiabeticComplicationsGlucoseSorbitolDesignofAldoseReductaseInhibitorsAldoseReductaseInhibitorHu&Chen,2003DatabaseforFunctionalGroupsLogIC50:0.6382,1.0LogIC50:0.6861,0.88Prediction:DrugLeadsStructure-activity-relationLogIC50:0.77,1.1LogIC50:-1.87,4.05LogIC50:-2.77,4.14LogIC50:0.68,0.88PredictionResultsusingAutoDockHu&Chen,2003Computer-aideddrugdesignChemicalSynthesisScreeningusinginvitroassayAnimalTestsClinicalTrialsBioinformaticsImprovecontent&utilityofbio-databasesDeveloptoolsfordatageneration,capture&annotationDeveloptoolsforcomprehensivefunctionalstudiesDeveloptoolsforrepresenting&analyzingsequencesimilarity&variationComputationalChemistryIncreasinglyimportantfieldinchemistryHelptounderstandexperimentalresultsProvideguidelinestoexperimentistsApplicationinMaterials&PharmaceuticalindustriesFuture:simulatenano-sizematerials,bulkmaterials;replaceexperimentalR&DObjective:Moreandmoreresearch&developmenttobeperformedoncomputersandInternetinsteadinthelaboratoriesQuantumChemistryG.H.ChenDepartmentofChemistryUniversityofHongKongContributors:Hartree,Fock,Slater,Hund,Mulliken,Lennard-Jones,Heitler,London,Brillouin,Koopmans,Pople,KohnApplication:Chemistry,CondensedMatterPhysics,MolecularBiology,MaterialsScience,DrugDiscoveryEmphasisHartree-FockmethodConceptsHands-onexperienceTextBook“QuantumChemistry”,4thEd.IraN.Levinehttp:/yangtze.hku.hk/lecture/chem3504-3.pptContents1.VariationMethod2.Hartree-FockSelf-ConsistentFieldMethod3.PerturbationTheory4.SemiempiricalMethodsTheVariationMethodConsiderasystemwhoseHamiltonianoperatorHistimeindependentandwhoselowest-energyeigenvalueisE1.Ifisanynormalized,well-behavedfunctionthatsatisfiestheboundaryconditionsoftheproblem,then *H dt E1ThevariationtheoremProof:Expand inthebasissetk=kkkwherekarecoefficientsHk=Ekkthen*H dt =kjk*jEjkj=k|k|2EkE1k|k|2=E1Sinceisnormalized,*dt =k|k|2=1i.:trialfunctionisusedtoevaluatetheupperlimitofgroundstateenergyE1ii.=groundstatewavefunction,*H dt=t=E1iii.optimizeparamemtersin byminimizing *H dt /t /*dt tRequirementsforthetrialwavefunction:i.zeroatboundary;ii.smoothnessamaximuminthecenter.Trialwavefunction:=x(l-x)Applicationtoaparticleinaboxofinfinitedepth0l *H dx=-(h2/8 2m)(lx-x2)d2(lx-x2)/dx2dx=h2/(4 2m)(x2-lx)dx=h2l3/(24 2m)*dx=x2(l-x)2dx=l5/30E=5h2/(4 2l2m)h2/(8ml2)=E1(1)Constructawavefunction(c1,c2,cm)(2)Calculatetheenergyof:EE(c1,c2,cm)(3)Choosecj*(i=1,2,m)sothatEisminimumVariationalMethodExample:one-dimensionalharmonicoscillatorPotential:V(x)=(1/2)kx2=(1/2)m 2x2=2 2m 2x2Trialwavefunctionforthegroundstate:(x)=exp(-cx2)*H dx=-(h2/8 2m)exp(-cx2)d2exp(-cx2)/dx2dx+2 2m 2 x2exp(-2cx2)dx=(h2/4 2m)(c/8)1/2+2m 2(/8c3)1/2 *dx=exp(-2cx2)dx=(/2)1/2c-1/2E=W=(h2/8 2m)c+(2/2)m 2/cTominimizeW,0=dW/dc=h2/82m-(2/2)m2c-2c=22m/hW=(1/2)h.E33E22E11ExtensionofVariationMethodForawavefunctionwhichisorthogonaltothegroundstatewavefunction1,i.e.dt*1=0E=dt*H/dt*E2thefirstexcitedstateenergyThetrialwavefunction:dt*1=0=k=1akkdt*1=|a1|2=0E=dt*H/dt*=k=2|ak|2Ek/k=2|ak|2k=2|ak|2E2/k=2|ak|2=E2e+1 2=c1 1+c2 2W=*H dt/*dt =(c12H11+2c1c2H12+c22H22)/(c12+2c1c2S+c22)W(c12+2c1c2S+c22)=c12H11+2c1c2H12+c22H22ApplicationtoH2+Partialderivativewithrespecttoc1(W/c1=0):W(c1+Sc2)=c1H11+c2H12Partialderivativewithrespecttoc2(W/c2=0):W(Sc1+c2)=c1H12+c2H22(H11-W)c1+(H12-SW)c2=0(H12-SW)c1+(H22-W)c2=0Tohavenontrivialsolution:H11-WH12-SWH12-SWH22-WForH2+,H11=H22;H120.GroundState:Eg=W1=(H11+H12)/(1+S)1=(1+2)/2(1+S)1/2ExcitedState:Ee=W2=(H11-H12)/(1-S)2=(1-2)/2(1-S)1/2=0bondingorbitalAnti-bondingorbitalResults:De=1.76eV,Re=1.32AExact:De=2.79eV,Re=1.06A1eV=23.0605kcal/molTrialwavefunction:k3/2-1/2exp(-kr)Eg=W1(k,R)ateachR,chooseksothatW1/k=0Results:De=2.36eV,Re=1.06AResutls:De=2.73eV,Re=1.06A1s2pInclusionofotheratomicorbitalsFurtherImprovementsH-1/2exp(-r)He+23/2-1/2exp(-2r)Optimizationof1sorbitalsa11x1+a12x2=b1a21x1+a22x2=b2(a11a22-a12a21)x1=b1a22-b2a12(a11a22-a12a21)x2=b2a11-b1a21LinearEquations1.twolinearequationsfortwounknown,x1andx2Introducingdeterminant:a11a12=a11a22-a12a21a21a22a11a12b1a12x1=a21a22b2a22a11a12a11b1x2=a21a22a21b2Ourcase:b1=b2=0,homogeneous1.trivialsolution:x1=x2=02.nontrivialsolution:a11a12=0a21a22nlinearequationsfornunknownvariablesa11x1+a12x2+.+a1nxn=b1a21x1+a22x2+.+a2nxn=b2.an1x1+an2x2+.+annxn=bna11a12.a1,k-1b1a1,k+1.a1na21a22.a2,k-1b2a2,k+1.a2ndet(aij)xk=.an1an2.an,k-1b2an,k+1.annwhere,a11a12.a1na21a22.a2ndet(aij)=.an1an2.anna11a12.a1,k-1b1a1,k+1.a1na21a22.a2,k-1b2a2,k+1.a2n.an1an2.an,k-1b2an,k+1.annxk=det(aij)inhomogeneouscase:bk=0foratleastonek(a)travialcase:xk=0,k=1,2,.,n(b)nontravialcase:det(aij)=0homogeneouscase:bk=0,k=1,2,.,nForan-thorderdeterminant,ndet(aij)=alkClkl=1where,ClkiscalledcofactorTrialwavefunctionisavariationfunctionwhichisacombinationofnlinearindependentfunctionsf1,f2,.fn,=c1f1+c2f2+.+cnfnn(Hik-SikW)ck=0i=1,2,.,nk=1SikdtfifkHikdtfiHfkWdt H/dt (i)W1W2.WnarenrootsofEq.(1),(ii)E1E2.EnEn+1.areenergiesofeigenstates;then,W1E1,W2E2,.,WnEnLinearvariationaltheoremMolecularOrbital(MO):=c11+c22(H11-W)c1+(H12-SW)c2=0S11=1(H21-SW)c1+(H22-W)c2=0S22=1Generally:iasetofatomicorbitals,basissetLCAO-MO=c1 1+c2 2+.+cn nlinearcombinationofatomicorbitalsn(Hik-SikW)ck=0i=1,2,.,nk=1Hikdti*HkSikdti*k Skk=1HamiltonianH=(-h2/2m)2 2-(-(h2/2me)i i2+Z Z e2 2/r-i Z Z e2/ri +i j e2/rijH(ri;r)=)=E E (ri;r)TheBorn-OppenheimerApproximation(1)(1)(ri;r)=)=el(ri;r)N(r)(2)H Hel(r)=-(-(h2/2me)i i2-i Z Z e2/ri +i j e2/rijVNN=Z Z e2 2/rHel(r)el(ri;r)=)=E Eel(r)el(ri;r)(3)HN=(-h2/2m)2 2+U(r)U(r)=E Eel(r)+VNNHN(r)N(r)=)=E E N(r)TheBorn-OppenheimerApproximation:AssignmentCalculatethegroundstateenergyandbondlengthofH2usingtheHyperChemwiththe6-31G(Hint:Born-OppenheimerApproximation)e+etwoelectronscannotbeinthesamestate.HydrogenMoleculeH2ThePauliprincipleSincetwowavefunctionsthatcorrespondtothesamestatecandifferatmostbyaconstantfactor(1,2)(1,2)=c2(2,1)(2,1)j ja(1)j(1)jb(2)+(2)+c1j ja(2)j(2)jb(1)(1)=c2j ja(2)j(2)jb(1)(1)+c2c1j ja(1)j(1)jb(2)(2)c1=c2c2c1=1Therefore:c1=c2=1AccordingtothePauliprinciple,c1=c2=-1Wavefunction:(1,2)(1,2)=j ja(1)j(1)jb(2)(2)+c1 j ja(2)j(2)jb(1)(1)(2,1)(2,1)=j ja(2)j(2)jb(1)(1)+c1 j ja(1)j(1)jb(2)(2)Wavefunction ofH2:(1,2)=1/(1,2)=1/2 2!(1)(1)(2)(2)-(2)(2)(1)(1)(1)(1)(2)(2)-(2)(2)(1)(1)(1)(1)(2)(2)(1)(1)(2)(2)=1/=1/2 2!(1)(1)(2)(2)(1)(1)(2)(2)ThePauliprinciple(differentversion)thewavefunctionofasystemofelectronsmustbeantisymmetricwithrespecttointerchangingofanytwoelectrons.SlaterDeterminantE=2 dt t1 1 *(1)(1)(Te+VeN)(1)(1)+VNN+dt t1 1 dt t2 2|2 2(1)|(1)|e2/r12|2 2(2)|(2)|=i=1,2fii+J12+VNNTominimizeE undertheconstraintdt|2|=1,useLagrangesmethod:L=E -2 dt1|2(1)|-1L=E -4 dt1*(1)(1)=4dt1*(1)(Te+VeN)(1)+4dt1 dt2*(1)*(2)e2/r12(2)(1)-4 dt1*(1)(1)=0Energy:E Te+VeN+dt t2 2 *(2)(2)e2/r12(2)(1)=e(1)(2)(1)=e(1)(f+J)=e )=e f(1)=Te(1)+VeN(1)oneelectronoperatorJ(1)=dt t2 2 *(2)(2)e2/r12(2)(2)twoelectronCoulomboperatorAverageHamiltonianHartree-Fockequationf(1)istheHamiltonianofelectron1intheabsenceofelectron2;J(1)isthemeanCoulombrepulsionexertedonelectron1by2;e e istheenergyoforbital.LCAO-MO:=c1 1+c2 2Multiple1fromtheleftandthenintegrate:c1F11+c2F12=(c1+Sc2)Multiple2fromtheleftandthenintegrate:c1F12+c2F22=(Sc1+c2)where,Fij=dt i*(f+J)j=Hij+dt i*JjS=dt 12(F11-)c1+(F12-S)c2=0(F12-S)c1+(F22-)c2=0SecularEquation:F11-e eF12-Se e =0=0F12-Se eF22-e e bondingorbital:1=(F11+F12)/(1+S)1 1 =(1 1+2 2)/2(1+S)1/2antibondingorbital:2=(F11-F12)/(1-S)2 2 =(1 1-2 2)/2(1-S)1/2MolecularOrbitalConfigurationsofHomonuclearDiatomicMoleculesH2,Li2,O,He2,etcMoeculeBondorderDe/eVH2+2.79H214.75He2+1.08He200.0009Li211.07Be200.10C226.3N2+8.85N239.91O2+26.78O225.21ThemoretheBondOrderis,thestrongerthechemicalbondis.BondOrder:one-halfthedifferencebetweenthenumberofbondingandantibondingelectrons-1-2 1(1)(1)(1)(1)2(1)(1)(1)(1)(1,2)=1/(1,2)=1/2 2 1(2)(2)(2)(2)2(2)(2)(2)(2)=1/2 1(1)(1)2(2)-(2)-2(1)(1)1(2)(1)(2)(2)(1)(2)E E =dt t1 1dt t2 2 *H =dt t1 1dt t2 2 *(T1+V1N+T2+V2N+V12+VNN)=(1)+(2)+(2)-(2)+VNN=i(1)+(2)-(2)+VNN=i=1,2fii+J12-K12+VNNParticleOne:f(1)+J2(1)-K2(1)ParticleTwo:f(2)+J1(2)-K1(2)f(j)-(-(h2/2me)j2-Z/rj Jj(1)q(1)q(1)q(1)q(1)dr2 2 j*(2)(2)e2/r12 j(2)(2)Kj(1)q(1)q(1)j(1)(1)dr2 2 j*(2)(2)e2/r12 q(2)q(2)AverageHamiltonian f(1)+J2(1)-K2(1)1(1)=1 1(1)f(2)+J1(2)-K1(2)2(2)=2 2(2)F(1)f(1)+J2(1)-K2(1)Fockoperatorfor1F(2)f(2)+J1(2)-K1(2)Fockoperatorfor2Hartree-FockEquation:FockOperator:1.AttheHartree-FockLeveltherearetwopossibleCoulombintegralscontributingtheenergybetweentwoelectronsiandj:CoulombintegralsJijandexchangeintegralKij;2.Fortwoelectronswithdifferentspins,thereisonlyCoulombintegralJij;3.Fortwoelectronswiththesamespins,bothCoulombandexchangeintegralsexist.Summary4.TotalHartree-Fockenergyconsistsofthecontributionsfromone-electronintegralsfiiandtwo-electronCoulombintegralsJijandexchangeintegralsKij;5.AttheHartree-FockLeveltherearetwopossibleCoulombpotentials(oroperators)betweentwoelectronsiandj:Coulomboperatorandexchangeoperator;Jj(i)istheCoulombpotential(operator)thatifeelsfromj,andKj(i)istheexchangepotential(operator)thatthatifeelsfromj.6.Fockoperator(or,averageHamiltonian)consistsofone-electronoperatorsf(i)andCoulomboperatorsJj(i)andexchangeoperatorsKj(i)N electronsspinupandN electronsspindown.Fockmatrixforanelectron1 withspinup:F(1)=f(1)+jJj(1)-Kj(1)+jJj(1)j=1,Nj=1,NFockmatrixforanelectron1 withspindown:F(1)=f(1)+jJj(1)-Kj(1)+jJj(1)j=1,N j=1,Nf(1)-(-(h2/2me)12-NZN/r1N Jj(1)dr2 2 j*(2)(2)e2/r12 j(2)(2)Kj(1)q(1)q(1)j(1)(1)dr2 2 j*(2)(2)e2/r12q(2)q(2)Energy=j fjj+j fjj+(1/2)i j(Jij-Kij)+(1/2)i j(Jij-Kij)+i j Jij +VNNi=1,N j=1,Nfjj fjj Jij Jij Kij Kij Jij Jij F(1)=f(1)+j=1,n/22Jj(1)-Kj(1)Energy=2 j=1,n/2fjj+i=1,n/2 j=1,n/2(2Jij-Kij)+VNNClosesubshellcase:(N=N=n/2)1.Many-BodyWaveFunctionisapproximatedbySlaterDeterminant2.Hartree-FockEquationFi=iiF Fockoperatorithei-thHartree-Fockorbitalitheenergyofthei-thHartree-FockorbitalHartree-FockMethod3.RoothaanMethod(introductionofBasisfunctions)i=kckikLCAO-MO kisasetofatomicorbitals(orbasisfunctions)4.Hartree-Fock-Roothaanequation j(Fij-e eiSij)cji=0Fij Sij 5.SolvetheHartree-Fock-Roothaanequationself-consistently )=)=(1)ifb=f,c=g,.,d=h;0,otherwise )=)=(2)ifc=g,.,d=h;0,otherwiseTheCondon-SlaterRules-thelowestunoccupiedmolecularorbital-thehighestoccupiedmolecularorbital-Theenergyrequiredtoremoveanelectronfromaclosed-shellatomormoleculesiswellapproximatedbyminustheorbitalenergyoftheAOorMOfromwhichtheelectronisremoved.HOMOLUMOKoopmansTheorem#HF/6-31G(d)RoutesectionwaterenergyTitle01MoleculeSpecificationO-0.4640.1770.0(inCartesiancoordinatesH-0.4641.1370.0H0.441-0.1430.0Slater-typeorbitals(STO)nlm=Nrn-1exp(-r/a0)Ylm(,)theorbital exponent*isusedinsteadofinthetextbookGaussiantypefunctionsgijk=Nxiyjzkexp(-ar2)(primitiveGaussianfunction)p=udupgu(contractedGaussian-typefunction,CGTF)u=ijkp=nlmBasisSet i=pcippBasissetofGTFsSTO-3G,3-21G,4-31G,6-31G,6-31G*,6-31G*-complexity&accuracyMinimalbasisset:oneSTOforeachatomicorbital(AO)STO-3G:3GTFsforeachatomicorbital3-21G:3GTFsforeachinnershellAO2CGTFs(w/2&1GTFs)foreachvalenceAO6-31G:6GTFsforeachinnershellAO2CGTFs(w/3&1GTFs)foreachvalenceAO6-31G*:addsasetofdorbitalstoatomsin2nd&3rdrows6-31G*:addsasetofdorbitalstoatomsin2nd&3rdrowsandasetofpfunctionstohydrogenPolarizationFunctionDiffuseBasisSets:Forexcitedstatesandinanionswhereelectronicdensityismorespreadout,additionalbasisfunctionsareneeded.Diffusefunctionsto6-31Gbasissetasfollows:6-31G*-addsasetofdiffuses&porbitalstoatomsin1st&2ndrows(Li-Cl).6-31G*-addsasetofdiffusesandporbitalstoatomsin1st&2ndrows(Li-Cl)andasetofdiffusesfunctionstoHDiffusefunctions+polarisationfunctions:6-31+G*,6-31+G*,6-31+G*and6-31+G*basissets.Double-zeta(DZ)basisset:twoSTOforeachAO6-31Gforacarbonatom:(10s4p)3s2p1s2s2pi(i=x,y,z)6GTFs 3GTFs 1GTF3GTFs1GTF1CGTF1CGTF1CGTF1CGTF1CGTF(s)(s)(s)(p)(p)Minimalbasisset:OneSTOforeachinner-shellandvalence-shellAOofeachatomexample:C2H2(2S1P/1S)C:1S,2S,2Px,2Py,2PzH:1Stotal12STOsasBasissetDouble-Zeta(DZ)basisset:twoSTOsforeachandvalence-shellAOofeachatomexample:C2H2(4S2P/2S)C:two1S,two2S,two2Px,two2Py,two2PzH:two1S(STOs)total24STOsasBasissetSplit-Valence(SV)bas