第三章1 majing print不穩(wěn)定容易得電子變?yōu)闅浞肿訉?shí)驗(yàn)證明它存在鍵長106pm

Chapter3DiatomicHydrogenMolecularIonMolecularHydrogenMolecularIonMolecularOrbitalMethod(MO)HydrogenMoleculeHomo-andHetero-NuclearDiatomicMoleculesValenceBondMethod(VB)ThenatureofchemicalOneothe hievementso uantummechanicsisitsabilitytodescribethechemicalbond.ReviewofChapter SchrdingerequationcannotbesolvedyticallywhenatomhasmorethanoneelectronApproximatenumericalmethodscanbeusedtoobtaintheeigenfunctionsandeigenvaluesoftheSchrdingerequationformanyelectronatoms Anewsetofquantumnumbersforthestatesofmany-electronatomsandthegrouofthesestatesintolevelsandtermswillbeconsiderate.2ThemolecularionH+involvesthesimplestchemical2H+鍵長106pm(2.00鍵解離能255.4kJ?mol-10.097Experimental H2Bond 106pm(2.00 255kJ/mol(0.097Short-lived,couldonlybedetectedbyM.S.thespark-dischargeprocessofhydrogen H1R2TheoreticalThree-bodyThreeparticles=>NineBorn-Oppenheimer1211

1 2e2e

rB 注意理解每一項(xiàng)的物理意義yPzAByPzABRe e

E B Exactsolutionofthisequation: Inthesameyearthehydrogen sasotreatedHeitlerandLondon

ValenceBondMethodMolecularOrbitalMethodTheAsymptoticBehavior R=一種可能 0Ep(Ep(R,----

EHEH1RABB B rBTheAsymptoticBehavior R= EHRABEp(Ep(R,----

TheAsymptoticBehavior R= Ep(R,Ep(R,AtomicLinearCombinationBLinearCombinationAtomicLinearCombinationB A

B

caAcaandcb:CombinationSymmetryintheSchroedingerySymmetryinHydrogenMoleculecacb=ca=cacb=ca=cb=c(AB c:Normalization22Wavefunction,normalization22c

d

Bd

Ad

d dS,OverlapA*B*A 2Wavefunction,BondingandAnti- 22S

1 Wavefunction,BondingandAnti- 吸引兩邊的原子核把它們拉在一起 (ABWavefunction,BondingandAnti- (A EnergyofMolecule?1211erArBKEnergyofMolecule *H?d *12111

B * 2 2

rB

1 d

1 2 A

R

r B

R rAB

*

21

B 1 d

*

21

B 1 A r B r B B 2

Symmetry(Aand 1EnergyofMoleculeWhentheelectronisclosetonucleusWhentheelectronisclosetonucleusA,thecoulombattractionbetweentheelectronandnucleusB.JJ1BEnergyofMolecule *

1 B

2

d

*

1AB BAB

AR

AB AAB?

KK*A1 rBBE1sS

EnergyofMolecule ) 1J 1JCoulombJ<0ExchangeK<0ComputationofJ,K,and J1e2R1 eR1RR2/EnergyCurveofBondingandAnti-BondingMolecular鍵長2.00鍵解離能0.097

E

E1s=-

BondLength(a0-0.5-(-0.565)=0.065a.u. BondingandAnti-At ondLength(2.5a0

1J 10.4a.u

J E1s

1SHydrogenHydrogenMoleculeDissociation10.4a.u

AtEquilibriumBondLength(2.5a02,5BasisBasisSet（基組iiIn2-D raxIn3-D raxbyInN-Dcc cInN-D Here,{x,y},{x,y,z}and{1, N}areorthonormalcompletesetin2-D,3-DandN-Dspaces,respectively.SimulatetheSimulatetherealCombinationsofmathematicalfunctionsusedtorepresentatomicorbitalsH: C,N,O:1s,2s,2px,2py,()()rexp(rtoodifficulttoyticallywhen–Gaussiantypeorbitalssimplertomanipulatemathematically;combinationsofGaussian(exp)functionscanapproximateSTO’sGaussiantypeorbitals–Linearcombinationsofgaussians;e.g.,STO-?3Gaussian“primitives”tosimulatea?(“Minimalbasis

True3gaussianrUsingthevariationalmethodtosolvetheSchr?dingerequationSchr?dingerequationforgroundstateWecanrewritetheequation theinte 0 ?0 Ifwecannotknowtheeigenfunction,howcanweapproachtoenergy(also,eigenvalue)?variational ?E

SetΦandoptimizationparameterα,wecanfindoptimum ipleoftheVariationForanarbitrarybasisset,N.Themolecularorbitalcouldbeexpandedtoitapproxima ciii

c11c22cNTheThevariationmethodistofindasetofcoefficients{ci}, aketheenergyexpectationvalue? closetoE0asEiEExamle articleina

2

(x)(x

x3)

x 7

9a

2a( d 2m

a

a

E ma02m(0

a3)(aa3Realminimumenergy:Trialenergyisbiggerthanrealenergyandtherearesmalldifferencebetweenrealwavefunctionandtrial2 d x 2m[(

a3)(

2( a9))]dx2[(aa3)(a52(a

a9Integrateit,wecanCalculatethedE/dα=0andfindextremevalueof→α=-5.74andα=-0.345,minimum:α=-Inthiscase,energyoftrialfunctionis0.127h2/ma2anditisveryclosetorealvalue.MethodologyoftheVariation *i?jd c *i?jd

H cic

H i1 *idcicjij cic*id

ijMethodologyoftheVariation cicj E j cicj jEMethodologyEnPartialdifferentiationofthisequationwithrespecttonnn

cicjHijESi1 (r1,2,,Nn n

E cc

1,2,,NMethodologyoftheVariationHomoenouseuationswithvariablesc1,c2 cN.Thenecessaryandsufficientconditionfornonzerosolutions.SecularEquation（久期方程SecularSolvetheequationtofindNeigenvaluesE1,E2,…,Thelowestvaluecorrespondto owest rgylevelSubstitutingEinthelinearhomogeneousequationwitheachEi,solvethisequationgivesasetof{ci},agoodapproximationoftheexpandingcoefficientsforcorrespondingwavefunction. H+的Schrdinger方程的變分求 c c c[ era]c[

erb)?(caacbb)d

c2H 2cc c 2cc cc 2cc c b將本征值代入久期方程，并用歸一化得本征函數(shù)LinearCombinationofAtomicOrbitalsi *

AtomicOrbitalsasbasisi= Coulomb

i Resonanceii

Overlapiijd iandjfromthesameatom,i0~ iandjfromdifferentCoulombi

H

Zr

l lr Zr l

ibelongskthZZr l

rdir

Z l AttractionsfromtheotherResonance

1

Z*d i r *d l lNon-classicalterm,alsocalledhointegralinsomephysicstextbooks,reflectsthecontributionofi-joverlaptothebondstrengthoftheinvolvedtwoatoms.SeeAppendixijji*SeeAppendixijji* H*ThePropertyofHAA

=HBAHBB=SAB=SBA=

Secular (1S2)E2( 2S)E( 2) 4(1S)( 2E

2S)24(1S2)( 2 ± IfIfBE+=hBBAhh 2ElectronicenergybeforeandafterbondBf E~After:E~2(A-Biggerhimplystrongerchemical( ( 22ABBA2 R =0,h=0 Wh