量子化学简介1113

量子化学简介1量子化学是用量子力学原理和方法研究分子微观结构以及与物质的物理和化学性质关系的一门边缘学科。在它过去多年的发展过程中，建立了一系列理论和计算方法，概括出许多对化学发展有重要意义的概念和规律。特别是七十年代以来，随着计算机技术的发展，量子化学已进入一个新的发展阶段，其研究成果使人们对物质结构与性质的内部联系以及化学现象本质与规律性的认识大大深化了。量子化学概述2理论方法从头算分子轨道理论（AbInitio

MolecularOrbitalTheory）半经验的分子轨道理论（Semi-EmpiricalMolecularOrbitalTheory）密度泛函理论（DensityFunctionalTheory)3Schrödingerequation: E=H

分子轨道理论4Schrödingerequation:kineticenergy(nuc.)kineticenergy(elect.)2kineticenergytermsplus3Coulombicenergyterms:(oneattractive,2repulsive)5SchrödingerequationafterBorn-OppenheimerApproximationkineticenergy(nuc.)kineticenergy(elect.)1kineticenergytermplus2Coulombicenergyterms:(oneattractive,1repulsive)plusaconstantfornuclei0constant6Simplifyingassumptionsareemployedto‘solve’theSchrödingerequationapproximately:Born-OppenheimerapproximationallowsseparatetreatmentofnucleiandelectronsHartree-Fockindependentelectronapproximationallowseachelectrontobeconsideredasbeingaffectedbythesum(field)ofallotherelectrons.LCAOApproximationBasisofM.O.Theory...7Born-OppenheimerApprox.Statesthatelectronmotionisindependentofnuclearmotion,thustheenergiesofthetwoareuncoupledandcanbecalculatedseparately.Derivesfromthelargedifferenceinthemassofnucleiandelectrons,andtheassumptionthatthemotionofnucleicanbeignoredbecausetheymoveveryslowlycomparedtoelectrons Htot

a(Tn)+Te+Vne+Vn+VeKineticenergyPotentialenergy(Tnisomitted;thisignoresrelativisticeffects,yieldingtheelectronicSchrödingerequation.)8Assumesthateachelectronexperiencesalltheothersonlyasawhole(fieldofcharge)ratherthanindividualelectron-electroninteractions.IntroducesaFockoperatorF: F whichisthesumofthekineticenergyofanelectron,apotentialthatoneelectronwouldexperienceforafixednucleus,andanaverageoftheeffectsoftheotherelectrons.Hartree-FockApproximation9LCAOApproximationElectronpositionsinmolecularorbitalscanbeapproximatedbyaLinearCombinationofAtomicOrbitals.Thisreducestheproblemoffindingthebestfunctionalformforthemolecularorbitalstothemuchsimpleroneofoptimizingasetofcoefficients(cn)inalinearequation:=c1

f1+c2

f2+c3

f3+c4

f4+…whereisthemolecularorbitalwavefunctionand

fn

representatomicorbitalwavefunctions.10BasissetsAbasissetisasetofmathematicalequationsusedtorepresenttheshapesofspaces(orbitals)occupiedbytheelectronsandtheirenergies.Basissetsincommonusehaveasimplemathematicalformforrepresentingtheradialdistributionofelectrondensity.MostcommonlyusedareGaussianbasissets,whichapproximatethebetter,butmorecomplicatedSlater-Typeorbitals(STO).11Computationalmethodology:guesstheorbitaloccupation(position)ofanelectronguessthepotentialeachelectronwouldexperiencefromallotherelectrons(takenasagroup)solveforFockoperatorstogenerateanew,improvedguessatthepositionsoftheelectronsrepeatabovetwostepsuntilthewavefunctionfortheelectronsisconsistentwiththefieldthatitandtheotherelectronsproduce(SCF).Hartree-FockSelf-ConsistentField(SCF)Method...12BasisSetsCombinationsofmathematicalfunctionsusedtorepresentatomicorbitalsMinimalH:1sC,N,O:1s,2s,2px,2py,2pz(allthreeneededtomaintainsphericalsymmetry)Slatertypeorbitals(STO)toodifficulttosolveanalyticallywhencombinedGaussiantypeorbitals(GTO)simplertomanipulatemathematically;combinationsofGaussian(exp)functionscanapproximateSTO’s13GaussianTypeOrbitalsSTO-3GSlatertypeorbitalsapproximatedbythreeprimitiveGaussianfunctionsUsedasdefaultbasissetinsemi-empiricalMOcalculations(AM1,PM3)BetterapproximationsusingcombinationsofGaussianfunctionshavebeendevelopedandaregenerallyemployedinabinitiowork14SplitBasisSetsMinimal(small)basissetssuchasSTO-3Gdonotadequatelydescribenon-spherical(anisotropic)electrondistributioninmolecules‘Split’valencebasissets(3-21G;6-31G,etc.)weredevelopedtoovercomethisproblemEachsplitvalenceatomicorbitaliscomposedofavariableproportionoftwo(ormore)functionsofdifferentsizeorradialextent15SplitBasisSets...3-21Gcommonlyusedsimplesplitbasisset;OKforHFgeometrycalculationson1strowelements,notgoodforheavierelementsorforaccurateenergies3primitiveGaussianfunctionsforinnercore(subvalence)electrons2Gaussiansforcontracted(small)valenceorbitals1Gaussianforextended(large)valenceorbitals16MoreSplitBasisSets...

withmodifications6-31G,6-311G(thelatterhastwodifferentsizesofextendedGaussianfunctionsforvalenceorbitals)Polarizationfunctions6-31G(d),or6-31G(d,p)[formerly6-31G*(or**)](adds‘d’functionto‘heavy’atoms,‘p’functiontoH,He)(aandbarecoefficientswhosesumis1)17MoreSplitBasisSets…

andstillmoremodificationsDiffusefunctions6-31+Gaddsanadditional,largerpfunctiontoheavy(non-hydrogenorhelium)atoms;indicatedby+beforeG6-31++Gaddsanadditional,largerpfunctiontoheavy(non-hydrogenorhelium)atomsandanadditionallargersfunctiontolightelements(hydrogenandhelium)Diffusefunctionsareusefulindescribinganions,moleculeswithlonepairsofe-,excitedstates,TS.18BasisSets:CommonCombinations6-31G(d) Common‘moderate’basisset6-31G(d,p)6-31+G(d,p)Goodcompromise6-31++G(d,p)Manyotherbasissetsareinuse,andbasissetscanbemodified/customized/optimizedeasily.19EffectofBasisSetChoiceonComputationCost(cputime)methylcyclohexaneonSGIIndigo2(Spartancputimeinsec.)

Method/BasisSet

s.p.

opt. AM1/STO-3G ~1 10 HF/STO-3G 72 983 HF/3-21G(d) 193 2214 HF/6-31G(d,p) 2632 34655(9.6h)(approaching“HFlimit”;energy[notshown]decreasesw/larger

basisset)

20SummaryofAbInitioMOTheoryGenerally,accuracyofresultsdependsonthedegreeofelectroncorrelationandthesizeofthebasissetused.Thecostofthecalculation(cputimerequired)increasesrapidlyasthebasissetsizeisincreasedandastheamountofelectroncorrelationincreases.Mostcalculationsrepresentacompromise.21Geometry(bondlengths,angles,dihedrals)Energy(enthalpyofformation,freeenergy)Vibrationalfrequencies,UV-VisspectraNMRchemicalshiftsIP,Electronaffinity(Koopman’stheorem)Atomicchargedistribution(...illdefined)Electrostaticpotential(interactionw/point+)Dipolemoment.PropertiesfromHartree-FockSelf-ConsistentField(SCF)Method...22Semi-Empirical

MolecularOrbitalTheoryUsessimplificationsoftheSchrödingerequationE=Htoestimatetheenergyofasystem(molecule)asafunctionofthegeometryandelectrondistribution.Thesimplificationsrequireempiricallyderived(nottheoretical)parameters(“fudgefactors”)tobringcalculatedvaluesinagreementwithobservedvalues,hencethetermsemi-empirical.231930’s Hückel treatedsystemsonly1952 Dewar PMO;firstsemi- quantitativeapplication1960’s Hoffmann ExtendedHuckel; includedbonds1965 Pople CNDO;firstusefulMO program1967 Pople INDO

HistoryofSemi-Empirical

MolecularOrbitalTheory241975 Dewar MINDO/3;waswidelyused1977 Dewar MNDO1985 Dewar AM1;addedvdW attraction&H-bonding1989 Stewart PM3;largertrainingset1970’s Zerner ZINDO;includestransition metals,parameterizedfor calculatingUV-VisspectraHistory...25Neglectcore(1s)electrons;replaceintegralforHcorebyanempiricalorcalculatedparameterNeglectvariousotherinteractionsbetweenelectronsonadjacentatoms:CNDO,INDO,MINDO/3,MNDO,etc.Addparameterssoastomakethesimplifiedcalculationgiveresultsinagreementwithobservables(spectraormolecularproperties).Semi-empiricalMOCalculations:

FurtherSimplifications26ConstructamodelorinputstructurefromMMcalculation,X-rayfile,orothersource(database)optimizestructureusingMMmethodtoobtainagoodstartinggeometryselectMOmethod(usuallyAM1orPM3)specifychargeandspinmultiplicity(s=n+1)selectsinglepointorgeometryoptimizationsetterminationcondition(time,cycles,gradient)selectkeywords(fromlistof>100).StepsinPerformingaSemi-empiricalMOCalculation27RelativeComputation“Cost”Molecularmechanics...cputimescalesassquareofthenumberofatoms...Calculationscanbeperformedonacompoundof~MW300inafewminutesonaPentiumcomputer,orinafewsecondsontheSGI.Thismeansthatlargermolecules(evenpeptides)andbemodeledbyMMmethods.28Relative