计算量子化学讲义

（理论部分）LectureNotesforComputationalQuantumChemistry2002年9月LecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•1•Chapter1PreparatoryKnowledgeofQuantumMechanics§1.1FundamentalPostulatesinQuantumMechanicsmatterparticleMicrocosmicparticlesm0=0:photonm00:matterparticles(electron,proton,neutron,atomsetc.)m0dualitydeBrogliedeBroglierelations:⎪⎩⎪⎨⎧==)()(momentumlinearnhpenergykinetichEvvlnEpvnlnvn,lnvNewtonLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•2•lPostulate1:4),,(where),,(zyxrtr=ΨvvN3N1N),,(where),,,,,(21iiiiNzyxrtrrr=ΨvvLvvN21±=Sm4N+1),,,(where),,,,,(21siiiiiNmzyxxtxxx=ΨvvLvv(1)free-particleinaninfinitespace)(),(EtpxiAetx-=Ψhone-dimensional)(),(EtrpiAetr-⋅=Ψvvhvthree-dimensional(1.1-1)(2)lparticlein1D-boxL,2,1,sin2)(==Ψnlxnlxnp(1.1-2)(3)1D-harmonicoscillatorhmwaaxxxx===Ψ-,),()(22xHeNnnn(1.1-3)Hn(x)Hermitepolynomial(4)),()(),,(,,,,jqjqmllnmlnYrRr=Ψ(1.1-4)Rn,l(r)LaguerreYl,m(q,j)TheStatisticalSignificationofY,Born,1926ΨEHΨΨ⇔E,H;I∝|Ψ|2LecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•3•OtdrvBornΨ|Ψ|2Ψ*Ψ),(trvΨtrvdtc|),(trvΨ|2dtc2|),(|),(trctrvvΨ=rtrv121222)|),(|(;1|),(|-∞∞∫∫Ψ=∴=Ψttdtrcdtrcvv(1.1-5)Ψ1|),(|2=Ψ∫∞tdtrvc=1normalized2|),(|),(trtrvvΨ=r(1.1-6)),(trvrlPostulate2:ObservablesarerepresentedbyHermiteoperatorLinearmomentum:vmpvv=Kineticenergy:mpmvT2||2122v==─1.DefinitionofHermiteoperatorFˆ()jytyjtjytjyandintegrableˆ**)ˆ(ˆ**∀==∫∫∫∞∞∞dFdFdF(1.1-7)Fˆself-adjointoperatorDiracDiracLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•4•∫〉〈=jytjy|ˆ|ˆ*FdF|j〉ket)(rvj〈y|bra)(*rvycomplexconjugate*)|(|〉=〈yy∫=〉〈tjyjyd*|Dirac(1.1-7)*|ˆ|*ˆ||ˆ|ˆ|〉〈=〉〈=〉〈=〉〈yjyjjyjyFFFF(1.1-8)dxdD=ˆdxdiDihh=ˆp2h=hhPlanck2.HowtoConstructanObservableOperator─rvpvt),,(tprfFvv=(1.1-9)wherepositionzkyjxirvvvv++=linearmomentumzyxpkpjpipvvvv++=FFˆ─rˆpˆt),ˆ,ˆ(ˆtprfF=wherepositionoperatorzkyjxirˆˆˆˆvvv++=linearmomentumoperatorzyxpkpjpipˆˆˆvvvv++=(1.1-10)uncertaintyprinciplerepresentationLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•5•positionrepresentationlinearmomentumrepresentationrrzzyyxxv====ˆ,ˆ,ˆ,ˆzipyipxipzyx∂∂-=∂∂-=∂∂-=hhhˆ,ˆ,ˆ∇-=⎟⎠⎞⎜⎝⎛∂∂+∂∂+∂∂-=hvvvhizkyjxiipˆzkyjxi∂∂+∂∂+∂∂=∇vvvgradiantoperatordelrvpˆFˆStep1:F),,(tprfFvv=Step2:pˆpv),ˆ,(ˆtprfFv=rvpˆ1-11-1mppmpT22||2vvv⋅==222)()(21ˆˆ21ˆ∇-=∇-⋅∇-=⋅=miimppmThhhwhere2222222zyx∂∂+∂∂+∂∂=∇Laplace),(trUUv=),(ˆtrUUUv==UTE+=),(2ˆ22trUmHvh+∇-=Hamiltonian3.EigenfunctionsandEigenvalueofOperatorsFˆfllff=Fˆ,or〉=〉flf||ˆF(1.1-11)fFˆl(1.1-11)LecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•6•1xeaadxdD=ˆaxxxeedxdeDaaaa)(ˆ==QaFˆL,2,1,||ˆ=〉=〉iFiiflf(1.1-12)Fˆ},2,1,{L=iifeigenfunctionsetofFˆlPostulate3:},2,1,{L=iifΨ∑=ΨiiicfPowerseries:∑∞==0)(kkkxaxfFouriesseries:()∑∞=++=10sincos2)(kkkkxbkxaaxf)(,!0∞=∑∞=|x|kxekkx)20(,)1()1(ln0≤--=∑∞=xkxxkkk3MOMOAOAOLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•7•lPostulate4:ΨFFFˆ〉〈ˆFexpectionvalue||ˆ|*ˆ*ˆ〉ΨΨ〈〉ΨΨ〈=ΨΨΨΨ=〉〈=∫∫FddFFFtt(1.1-13)1Ψ1|=〉ΨΨ〈〉ΨΨ〈=|ˆ|FF2ΨFˆl〉Ψ=〉Ψ||ˆlFl≡Fˆ3ΨFˆ〉〈ˆFlPostulate5:identicalparticles1.1(a)1122(a)(b)(c)1.1(a)(b)(c)[1.1(b)][1.1(c)]AOMOLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•8•§1.2EigenvaluesandEigenfunctionsofObservableOperatorsFˆfllyy=Fˆ〉=〉yly||ˆFfFˆl1.2.1EigenvaluesTheorem1:Theeigenvaluesofanhermitianoperatormustbereal.Proof:AssumethatFˆishermitianand〉=〉flf||ˆF,wehave〉〈〉〈=〉〈=〉〈fflfflffff||||*|ˆ|ˆ|FFItfollowsthat0|*)(=〉〈-ffll(1.2-1)Q0||*|2==〉〈∫∫∞∞tftffffdd,TheholdofEq.(1.2-1)demandsthat∴==-,*,0*lllllmustberealQ.E.D.1.2.2EigenfunctionsTheorem2:Twoeigenfunctionscorrespondingtodifferenteigenvaluesofanhermitianoperatorareorthogonalwitheachother.Proof:AssumethatFˆishermitianand〉=〉kkkFflf||ˆ,〉=〉lllFflf||ˆlkll≠wehave〉〈〉〈=〉〈=〉〈=〉〈lkllkklkklklkFFfflfflfflffff||||||ˆ|ˆ|*LecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•9•Itfollowsthat0|)(=〉〈-lklkffll(1.2-2)0|0=〉〈⇒≠-lklkffll,i.e.fkandflareorthogonalQ.E.D.l02l+1SchmidtJacobi(orthonormalcompleteset){yi,i=1,2,…}Kroneckd-ijjidyy=〉〈|,where⎩⎨⎧=∀≠∀=,1,0jijiijdisKroneckd-symbol§1.3SchrödingerEquationsofMolecularSystemsHˆΨSchrödingerF=maMOVBDFTSchrödinger1.3.1Time-DependentSchrödingerEquationHˆΨtSchrödingerΨ=∂Ψ∂Htiˆh,(1.3-1)where),(trvΨ=Ψ,),(ˆ),(ˆˆtrUTtrHHvv+==LecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•10•HˆTˆ),(trUv⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+∂∂-=22222222ˆzyxmThHˆU1.3.2Time-IndependentSchrödingerEquationU0),(=∂∂=tUrUUvwehave)(ˆˆ..,0ˆrHHeitHv==∂∂.ΨstandingwaveHˆΨSchrödingerSchrödingerstationarystateSchrödingerequationΨ=ΨEHˆ(1.3-2)E)(rvΨ=ΨSchrödingerSchrödingerrZemH22022ˆ-∇-=hm0eZr1.3.3StationaryStateSchrödingerEquationforMoleculesadiabaticLecturenotesforComputationalQuantumChemistryChap.1PreparatoryKnowledgeofQuant.Mech.•11•SchrödingerΨ=ΨEHˆ∑∑∑∑∑++-∇-∇-=iqppqqpijijippipiipppReZZrereZmMH22,2222222ˆhh(1.3-3)pqijMppmiiHˆsecJ10626.6where,232⋅×==-hhphplancke=1.602×10-19Cmim