量子化學與群論基礎6課件

6.3Many-electronatoms1TheSchr?dingerequationofmany-electronatoms（Born-Oppenheimer

Approximation）Unfortunately,precisesolutionsarenotavailablethroughtheSchr?dingerequation,evenforthesimplestmany-electron,helium,because6.3Many-electronatoms1The1

⑴IndependentparticlemodelTheSchr?dingerequationSeparationofvariables⑴IndependentparticlemodelT2

⑵MeanfieldmodelAnelectronatadistancerfromthenucleusexperiencesaCoulombicrepulsionfromalltheelectronswithinasphereofradiusrandwhichisequivalenttoapointnegativechargelocatedonthenucleus.，

n=1,2,3,……⑵MeanfieldmodelAnelectro3Symmetric,Bosons

Antisymmetric,Fermions

⑵ThePauliprinciple

Allelectronicwavefunctionsmustbeantisymmetricundertheinterchangeofanytwoelectrons.2IdenticalparticlesandthePauliprinciple⑴IdenticalparticlesIdenticalparticlescannotbedistinguishedbymeansofanyintrinsicproperties.Symmetric,BosonsAntisymmetri4⑶Slaterdeterminant—Normalizationconstant(i)(ii)Notwoelectronsinanatomcanhavethesamevaluesforallfourquantumnumbers.⑶Slaterdeterminant—Normaliz5

4Electronconfigurations⑴ThePauliexclusionprincipleNotwoelectronsinanatomcanhavethesamevaluesforallfourquantumnumbers.⑵Groundstateelectronconfiguration—Aufbauprinciple4Electronconfigurations⑴T6⑶Hund’sruleElectronsoccupytheorbitalsofasubshellsinglyuntileachorbitalhasoneelectron.p6,d10,f14

p3,d5,f7

p0,d0,f0⑶Hund’srulep6,d10,f14 7⑵Atomicunits1a.umass=themassofelectronm=9.109×1028g1a.ucharge=thechargeofprotone=1.602×10-19C1a.ulength=Bohrradius1a.uenergy=e2/a0=27.2eVTheH2+hastwoprotonsandoneelectronandcanbedescribedusingtheSchr?dingerequation5Molecules5.1Hydrogen

Molecule

Ion（H2+）⑴TheSchr?dingerequationofH2+⑵Atomicunits1a.umass=the8⑶TheSchr?dingerequationofH2+ina.u①TheHamiltoniana.u②Schr?dingerequation⑷Thevariationtheorem①Thevariationtheoremforalinearexpansion⑶TheSchr?dingerequationof9①TheestimatedwavefunctionTheestimatedwavefunctionhastosatisfysomeconditions.NotethatwehavetousethecorrectHamiltonianforthesystem,butwedonotknowhowtosolvetheSchr?dingerequationforthisHamiltonian.Thevariationtheoremtellsusthat:<E>ETheexpectationvalueoftheenergyisalwayshigherthanthecorrectresult.

MolecularOrbital--aLinearCombinationofAtomicOrbitalsLCAO-MO①Theestimatedwavefunction10②Expectationvalueoftheenergy〈E〉Theproblemisamaximum-minimumproblemincalculus.Wemusthave:③Thewavefunction⑸ThesolutionofSchrodingerequationofH2+②Expectationvalueoftheen11LCAO-MOR→∞,ra→∞,①TheestimatedwavefunctionIfR→∞,ra→∞,thenLCAO-MOR→∞,ra→∞,①Theesti12②TheenergyofH2+②TheenergyofH2+13Alltheintegralsabovecaninprinciplebeevaluated.Weknowthefunctionsandtheoperator.Wewilljustgivethemnames:soTheseequationsarecalledlinearhomogeneousequations.Alltheintegralsabovecanin14TheseculardeterminantHaa=Hbb,

Hab=Hba,

Sab=Sba,and

c1=c2

Theimportantquestioniswhetherthereisasolutionotherthanthetrivialsolution.Thereis.Thewavefunctiondisappears(thetrivialsolution)forallvaluesof<E>exceptforthevaluesof<E>thatsatisfythedeterminantequation:

c1=-c2TheseculardeterminantHaa=Hbb15③ApproximatewavefunctionsolvetheequationforE1NormalizationsolvetheequationforE2SoNormalization③Approximatewavefunctionsol16④TheintegralsSab,HaaandHab

（i）Sab—theoverlapintegralR0,soSab0.IfR=0,Sab=1;R=∞,Sab=0.④TheintegralsSab,HaaandH17

（ii）Haa—Coulombintegral（ii）Haa—Coulombintegral18(iii）Hab—exchangeintegral(integral)R>0，soHab<0,HabR↑，｜Hab｜↓，Sab<<1，E1=Haa+Hab=+

，E2=Haa-Hab=-

HaaEa，soE1=Ea+

，E2=Ea-

(iii）Hab—exchangeintegral(19⑤Discussion（i）Theenergyof1and2Thecalculatedandexperimentalmolecularpotentialenergycurvesforahydrogenmolecule-ion.（ii）Bondingorbital1⑤Discussion（i）Theenergyof20Theelectrondensitycalculatedbyformingthesquareofthewavefunction.Notetheaccumulationofelectrondensityintheinternuclearregion.Theboundarysurfaceofa(orbitalenclosestheregionwheretheelectronsthatoccupytheorbitalaremostlikelytobefound.Notethattheorbitalhascylindricalsymmetry.Theelectrondensitycalculate21（iii）Antibondingorbital2（iii）Antibondingorbital222Apartialexplanationoftheoriginofbondingandantibondingeffects.(a)Inabondingorbital,thenucleiareattractedtotheaccumulationofelectrondensityintheinternuclearregion.(b)Inanantibondingorbital,thenucleiareattractedtoanaccumulationofelectrondensityoutsidetheinternuclearregion.Apartialexplanationoftheo235.2.Molecularorbitaltheory(MOtheory)1.ThemolecularHamiltonianAmoleculeconsistsofnumberofelectronsandnuclei.ThemolecularHamiltonianoperator

hasacomplicatedform.

=(1,2,N)：(WithintheBorn--Oppenheimerapproximation)MainapproximationofabinitioMOtheory

theBorn--OppenheimerapproximationTheorbitalapproximationNon-relativityapproximation5.2.Molecularorbitaltheory242.Themolecularwavefunctions(molecularorbitals)Solet'sconsiderasimplerproblem,involvingtheone-electronhamiltonianSeparationofvariables(1,2,3…N)2.Themolecularwavefunction25(1,2,…,N)=det{(1)(1)(2)(2)…(N)(N)}3.Variationalparameter,orD=C?CDiscalledthedensitymatrix,aproductofAO--MOcoefficientmatrices(1,2,…,N)=det{(1)(1)(2)264.Hartree-FockequationsLetslookatageneralexampleoffunctionalvariationWritingtheenergyaswewantE=0,soThus4.Hartree-FockequationsLets27Itisclearthatthiscanbewrittenasamatrixproduct,andisinfactaneigenvalueequationintheform

Hc=ScEwecanrewritetheHartree-FockequationsasUsingthefactthat

isdiagonal,thiscanbewrittenasthematrixproduct

FC=SC

www.adi.uam.es\Docs\Knowledge\Fundamental_Theory\hf\hf.htmlItisclearthatthiscanbew28CapabilitiesofabinitioquantumchemistryCancalculatewavefunctionsanddetaileddescriptionsofmolecularorbitalsCancalculateatomiccharges,dipolemoments,multipolemoments,polarisabilities,etc.Cancalculatevibrationalfrequencies,IRandRamanintensities,NMRchemicalshiftsCancalculateionisationenergiesandelectronaffinitiesCanincludetheelectrostaticeffectsonsolvationCancalculatethegeometriesandenergiesofequilibriumstructures,transitionstructures,intermediates,andneutralandchargedspeciesCancalculategroundandexcitedstatesCanhandleanyelectronconfigurationCanhandleanyelementCanoptimisegeometriesCapabilitiesofabinitioquan295.3TheHuckelMoleculorOrbitalmethod(HMO)HMOdealwithconjugatedmolecules.Butadiene,e.g.：61s+4(1s22s22px12py12pz0)=26AOHMOapproximation:4pz.Inhisapproach,Theorbitalsaretreatedseparatelyfromtheorbitals,andthelatterformarigidframeworkthatdeterminetheshapeofthemolecule.⑴HuckelapproximationIHMOissuggestedbyEricHückelin1931.5.3TheHuckelMoleculorOrb30Butadiene4pzofCatomsButadiene4pzofCatoms31Theenergyandcoefficientssatisfythefollowingequations:let

Thebestmolecularorbitalsarethosewhichminimisethetotalenergy.Thisisachievedbyimposingthecondition:：Theenergyandcoefficientss32⑵HuckelapproximationII：non-trivialsolutions：Thesevalues,calledthenon-trivialsolutionstotheseequations,occurwhen:⑵HuckelapproximationII：no33letThisdeterminantcanbeeasilymultipliedouttogive:x4-3x2+1=0letThisdeterminantcanbeeas341=0.37171+0.60152+0.60153+0.371742=0.60151+0.37172—0.37173—0.601543=0.60151—0.37172—0.37173+0.601544=0.37171—0.60152+0.60153—0.37174<0,soE1<E2<E3<E4WeobtainfourvaluesofE,whichisreasonablesinceweexpecttofindfourmolecularorbitals.1=0.37171+0.60152+0.60153+35DelocalizationenergyTotalenergyE=2E1+2E2=2×(+1.62)+2×(+0.62)=4+4.48EnergylevelsOccupiedorbitalUnfilledorbitalC=C—C=CE’=4+4E-E’=0.48FrontierorbitalsThehighestoccupiedmolecularorbital,HOMOThelowestunfilledmolecularorbital,LUMOThefrontierorbitalsareimportantbecausetheyarelargelyresponsibleformanyofthechemicalandspectroscopicpropertiesofthemolecule.DelocalizationenergyEnergyl366.3Many-electronatoms1TheSchr?dingerequationofmany-electronatoms（Born-Oppenheimer

Approximation）Unfortunately,precisesolutionsarenotavailablethroughtheSchr?dingerequation,evenforthesimplestmany-electron,helium,because6.3Many-electronatoms1The37

⑴IndependentparticlemodelTheSchr?dingerequationSeparationofvariables⑴IndependentparticlemodelT38

⑵MeanfieldmodelAnelectronatadistancerfromthenucleusexperiencesaCoulombicrepulsionfromalltheelectronswithinasphereofradiusrandwhichisequivalenttoapointnegativechargelocatedonthenucleus.，

n=1,2,3,……⑵MeanfieldmodelAnelectro39Symmetric,Bosons

Antisymmetric,Fermions

⑵ThePauliprinciple

Allelectronicwavefunctionsmustbeantisymmetricundertheinterchangeofanytwoelectrons.2IdenticalparticlesandthePauliprinciple⑴IdenticalparticlesIdenticalparticlescannotbedistinguishedbymeansofanyintrinsicproperties.Symmetric,BosonsAntisymmetri40⑶Slaterdeterminant—Normalizationconstant(i)(ii)Notwoelectronsinanatomcanhavethesamevaluesforallfourquantumnumbers.⑶Slaterdeterminant—Normaliz41

4Electronconfigurations⑴ThePauliexclusionprincipleNotwoelectronsinanatomcanhavethesamevaluesforallfourquantumnumbers.⑵Groundstateelectronconfiguration—Aufbauprinciple4Electronconfigurations⑴T42⑶Hund’sruleElectronsoccupytheorbitalsofasubshellsinglyuntileachorbitalhasoneelectron.p6,d10,f14

p3,d5,f7

p0,d0,f0⑶Hund’srulep6,d10,f14 43⑵Atomicunits1a.umass=themassofelectronm=9.109×1028g1a.ucharge=thechargeofprotone=1.602×10-19C1a.ulength=Bohrradius1a.uenergy=e2/a0=27.2eVTheH2+hastwoprotonsandoneelectronandcanbedescribedusingtheSchr?dingerequation5Molecules5.1Hydrogen

Molecule

Ion（H2+）⑴TheSchr?dingerequationofH2+⑵Atomicunits1a.umass=the44⑶TheSchr?dingerequationofH2+ina.u①TheHamiltoniana.u②Schr?dingerequation⑷Thevariationtheorem①Thevariationtheoremforalinearexpansion⑶TheSchr?dingerequationof45①TheestimatedwavefunctionTheestimatedwavefunctionhastosatisfysomeconditions.NotethatwehavetousethecorrectHamiltonianforthesystem,butwedonotknowhowtosolvetheSchr?dingerequationforthisHamiltonian.Thevariationtheoremtellsusthat:<E>ETheexpectationvalueoftheenergyisalwayshigherthanthecorrectresult.

MolecularOrbital--aLinearCombinationofAtomicOrbitalsLCAO-MO①Theestimatedwavefunction46②Expectationvalueoftheenergy〈E〉Theproblemisamaximum-minimumproblemincalculus.Wemusthave:③Thewavefunction⑸ThesolutionofSchrodingerequationofH2+②Expectationvalueoftheen47LCAO-MOR→∞,ra→∞,①TheestimatedwavefunctionIfR→∞,ra→∞,thenLCAO-MOR→∞,ra→∞,①Theesti48②TheenergyofH2+②TheenergyofH2+49Alltheintegralsabovecaninprinciplebeevaluated.Weknowthefunctionsandtheoperator.Wewilljustgivethemnames:soTheseequationsarecalledlinearhomogeneousequations.Alltheintegralsabovecanin50TheseculardeterminantHaa=Hbb,

Hab=Hba,

Sab=Sba,and

c1=c2

Theimportantquestioniswhetherthereisasolutionotherthanthetrivialsolution.Thereis.Thewavefunctiondisappears(thetrivialsolution)forallvaluesof<E>exceptforthevaluesof<E>thatsatisfythedeterminantequation:

c1=-c2TheseculardeterminantHaa=Hbb51③ApproximatewavefunctionsolvetheequationforE1NormalizationsolvetheequationforE2SoNormalization③Approximatewavefunctionsol52④TheintegralsSab,HaaandHab

（i）Sab—theoverlapintegralR0,soSab0.IfR=0,Sab=1;R=∞,Sab=0.④TheintegralsSab,HaaandH53

（ii）Haa—Coulombintegral（ii）Haa—Coulombintegral54(iii）Hab—exchangeintegral(integral)R>0，soHab<0,HabR↑，｜Hab｜↓，Sab<<1，E1=Haa+Hab=+

，E2=Haa-Hab=-

HaaEa，soE1=Ea+

，E2=Ea-

(iii）Hab—exchangeintegral(55⑤Discussion（i）Theenergyof1and2Thecalculatedandexperimentalmolecularpotentialenergycurvesforahydrogenmolecule-ion.（ii）Bondingorbital1⑤Discussion（i）Theenergyof56Theelectrondensitycalculatedbyformingthesquareofthewavefunction.Notetheaccumulationofelectrondensityintheinternuclearregion.Theboundarysurfaceofa(orbitalenclosestheregionwheretheelectronsthatoccupytheorbitalaremostlikelytobefound.Notethattheorbitalhascylindricalsymmetry.Theelectrondensitycalculate57（iii）Antibondingorbital2（iii）Antibondingorbital258Apartialexplanationoftheoriginofbondingandantibondingeffects.(a)Inabondingorbital,thenucleiareattractedtotheaccumulationofelectrondensityintheinternuclearregion.(b)Inanantibondingorbital,thenucleiareattractedtoanaccumulationofelectrondensityoutsidetheinternuclearregion.Apartialexplanationoftheo595.2.Molecularorbitaltheory(MOtheory)1.ThemolecularHamiltonianAmoleculeconsistsofnumberofelectronsandnuclei.ThemolecularHamiltonianoperator

hasacomplicatedform.

=(1,2,N)：(WithintheBorn--Oppenheimerapproximation)MainapproximationofabinitioMOtheory

theBorn--OppenheimerapproximationTheorbitalapproximationNon-relativityapproximation5.2.Molecularorbitaltheory602.Themolecularwavefunctions(molecularorbitals)Solet'sconsiderasimplerproblem,involvingtheone-electronhamiltonianSeparationofvariables(1,2,3…N)2.Themolecularwavefunction61(1,2,…,N)=det{(1)(1)(2)(2)…(N)(N)}3.Variationalparameter,orD=C?CDiscalledthedensitymatrix,aproductofAO--MOcoefficientmatrices(1,2,…,N)=det{(1)(1)(2)624.Hartree-FockequationsLetslookatageneralexampleoffunctionalvariationWritingtheenergyaswewantE=0,soThus4.Hartree-FockequationsLets63Itisclearthatthiscanbewrittenasamatrixproduct,andisinfactaneigenvalueequationintheform

Hc=ScEwecanrewritetheHartree-FockequationsasUsingthefactthat

isdiagonal,thiscanbewrittenasthematrixproduct

FC=SC

www.adi.uam.es\Docs\Knowledge\Fundamental_Theory\hf\hf.htmlItisclearthatthiscanbew64CapabilitiesofabinitioquantumchemistryCancalculatewavefunctionsanddetaileddescriptionsofmolecularorbitalsCancalculateatomiccharges,dipolemoments,multipolemoments,polarisabilities,etc.Cancalculatevibrationalfrequencies,IRandRamanintensities,NMRchemicalshiftsCancalculateionisationenergiesandelectronaffinitiesCanincludetheelectrostaticeffectsonsolvationCancalculatethegeometriesandenergiesofequilibriumstructures,transitionstructures,intermediates,andneutralandchargedspeciesCancalculategroundandexcitedstatesCanhandleanyelectronconfigurationCanhandleanyelementCanoptimisegeometriesCapabilitiesofabiniti