量子化學(xué)簡介1量子化學(xué)是用量子力學(xué)原理和方法研究分子微觀結(jié)構(gòu)以及與物質(zhì)的物理和化學(xué)性質(zhì)關(guān)系的一門邊緣學(xué)科。在它過去多年的發(fā)展過程中，建立了一系列理論和計算方法，概括出許多對化學(xué)發(fā)展有重要意義的概念和規(guī)律。特別是七十年代以來，隨著計算機(jī)技術(shù)的發(fā)展，量子化學(xué)已進(jìn)入一個新的發(fā)展階段，其研究成果使人們對物質(zhì)結(jié)構(gòu)與性質(zhì)的內(nèi)部聯(lián)系以及化學(xué)現(xiàn)象本質(zhì)與規(guī)律性的認(rèn)識大大深化了。量子化學(xué)概述2理論方法從頭算分子軌道理論（AbInitio

MolecularOrbitalTheory）半經(jīng)驗的分子軌道理論（Semi-EmpiricalMolecularOrbitalTheory）密度泛函理論（DensityFunctionalTheory)3Schr?dingerequation: E=H

分子軌道理論4Schr?dingerequation:kineticenergy(nuc.)kineticenergy(elect.)2kineticenergytermsplus3Coulombicenergyterms:(oneattractive,2repulsive)5Schr?dingerequationafterBorn-OppenheimerApproximationkineticenergy(nuc.)kineticenergy(elect.)1kineticenergytermplus2Coulombicenergyterms:(oneattractive,1repulsive)plusaconstantfornuclei0constant6Simplifyingassumptionsareemployedto‘solve’theSchr?dingerequationapproximately:Born-OppenheimerapproximationallowsseparatetreatmentofnucleiandelectronsHartree-Fockindependentelectronapproximationallowseachelectrontobeconsideredasbeingaffectedbythesum(field)ofallotherelectrons.LCAOApproximationBasisofM.O.Theory...7Born-OppenheimerApprox.Statesthatelectronmotionisindependentofnuclearmotion,thustheenergiesofthetwoareuncoupledandcanbecalculatedseparately.Derivesfromthelargedifferenceinthemassofnucleiandelectrons,andtheassumptionthatthemotionofnucleicanbeignoredbecausetheymoveveryslowlycomparedtoelectrons Htot

a(Tn)+Te+Vne+Vn+VeKineticenergyPotentialenergy(Tnisomitted;thisignoresrelativisticeffects,yieldingtheelectronicSchr?dingerequation.)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

MolecularOrbitalTheoryUsessimplificationsoftheSchr?dingerequationE=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