香港大學(xué)量子化學(xué)

ComputationalChemistryG.H.CHENDepartmentofChemistryUniversityofHongKongIn1929,Diracdeclared,“Theunderlyingphysicallawsnecessaryforthemathematicaltheoryof...thewholeofchemistryarethuscompletelyknow,andthedifficultyisonlythattheexactapplicationoftheselawsleadstoequationsmuchtoocomplicatedtobesoluble.”BeginningofComputationalChemistryTheoryisreality! W.A.GoddardIIIDiracComputationalChemistryQuantumChemistryMolecularMechanicsBioinformatics

Create&AnalyseBio-information

Schr?dingerEquationF=MaMulliken,1966Fukui,1981Hoffmann,1981Pople,1998Kohn,1998NobelPrizesforComputationalChemsitryComputationalChemistryIndustryCompanySoftwareGaussianInc. Gaussian94,Gaussian98Schr?dingerInc. JaguarWavefunction SpartanQ-Chem Q-ChemAccelrys InsightII,Cerius2HyperCube HyperChemCeleraGenomics(Dr.CraigVenter,formalProf.,SUNY,Baffalo;98-01)Applications:materialdiscovery,drugdesign&researchR&DinChemical&Pharmaceuticalindustriesin2000:US$80billionBioinformatics:TotalSalesin2001 US$225million ProjectSalesin2006 US$1.7billionLODESTARv1.02--LocalizedDensityMatrix:STARperformerhttp://yangtze.hku.hkSoftwareDevelopmentatHKUQuantumChemistryMethodsAbinitiomolecularorbitalmethodsSemiempiricalmolecularorbitalmethodsDensityfunctionalmethodH

y=E

ySchr?dingerEquationHamiltonianH=

(-h2/2ma)

2-(h2/2me)

i

i2

+

ZaZbe2/rab-

i

Zae2/ria

+

i

j

e2/rijWavefunctionEnergyVitaminCC60CytochromechemeOH+D2-->HOD+D

energyC60andSuperconductorApplications:Magnet,Magnetictrain,PowertransportationWhatissuperconductor?ElectricalCurrentflowsforever!SoccerBallCrystalStructureofC60solidCrystalStructureofK3C60

K3C60isaSuperconductor(Tc=19K)Erwin&Pickett,Science,1991GHChen,Ph.D.Thesis,Caltech(1992)VibrationSpectrumofK3C60EffectiveAttraction!VibrationofAtomsThemechanismofsuperconductivityinK3C60wasdiscoveredusingcom-putationalchemistrymethodsVarmaet.al.,1991;Schluteret.al.,1992;Dresselhauset.al.,1992;Chen&Goddard,1992CarbonNanotubes(Ijima,1991)STMImageofCarbonNanotubes(Wildoeret.al.,1998)CalculatedSTMImageofaCarbonNanotube(Rubio,1999)ComputerSimulations(Saito,Dresselhaus,Louieet.al.,1992)CarbonNanotubes(n,m): Conductor, ifn-m=3II=0,±1,±2,±3,…;or Semiconductor,ifn-m

3IMetallicCarbonNanotubes: ConductingWiresSemiconductingNanotubes: TransistorsMolecular-scalecircuits! 1nmtransistor!0.13μmtransistor!30nmtransistor!Wildoer,Venema,Rinzler,Smalley,Dekker,Nature391,59(1998)ExperimentalConfirmations:Lieberet.al.1993;Dravidet.al.,1993;Iijimaet.al.1993;Smalleyet.al.1998;Haddonet.al.1998;Liuet.al.1999Science9thNovember,2001Logicgates(andcircuits)withcarbonnanotucetransistorScience7thJuly,2000Carbonnanotube-BasednonvolatileRAMformolecularcomputingNanoelectromechanicalSystems(NEMS)K.E.Drexler,Nanosystems:MolecularMachinery,ManufacturingandComputation(Wiley,NewYork,1992).LargeGearDrivesSmallGearG.Honget.al.,1999Nano-oscillatorsZhao,Ma,Chen&Jiang,Phys.Rev.Lett.2003NanoscopicElectromechanicalDevice (NEMS)HibernationAwakeningOscillationQuantummechanicalinvestigationofthefieldemissionfromthetipsofcarbonnanotubesZettl,PRL2001Zheng,Chen,Li,Deng&Xu,Phys.Rev.Lett.2004Computer-AidedDrugDesignGENOMICSHumanGenomeProjectDrugDiscoveryALDOSEREDUCTASEDiabetesDiabeticComplicationsGlucoseSorbitolDesignofAldoseReductaseInhibitorsAldoseReductaseInhibitorHu&Chen,2003DatabaseforFunctionalGroupsLogIC50:0.6382,1.0LogIC50:0.6861,0.88

Prediction:DrugLeadsStructure-activity-relationLogIC50:0.77,1.1LogIC50:-1.87,4.05LogIC50:-2.77,4.14LogIC50:0.68,0.88PredictionResultsusingAutoDockHu&Chen,2003Computer-aideddrugdesignChemicalSynthesisScreeningusinginvitroassayAnimalTestsClinicalTrialsBioinformaticsImprovecontent&utilityofbio-databasesDeveloptoolsfordatageneration,capture&annotationDeveloptoolsforcomprehensivefunctionalstudiesDeveloptoolsforrepresenting&analyzingsequencesimilarity&variationComputationalChemistryIncreasinglyimportantfieldinchemistryHelptounderstandexperimentalresultsProvideguidelinestoexperimentistsApplicationinMaterials&PharmaceuticalindustriesFuture:simulatenano-sizematerials,bulkmaterials;replaceexperimentalR&DObjective:Moreandmoreresearch&developmenttobeperformedoncomputersandInternetinsteadinthelaboratoriesQuantumChemistryG.H.ChenDepartmentofChemistryUniversityofHongKongContributors:

Hartree,Fock,Slater,Hund,Mulliken,Lennard-Jones,Heitler,London,Brillouin,Koopmans,Pople,KohnApplication:

Chemistry,CondensedMatterPhysics,MolecularBiology,MaterialsScience,DrugDiscoveryEmphasis

Hartree-Fockmethod Concepts Hands-onexperienceTextBook

“QuantumChemistry”,4thEd. IraN.Levinehttp://yangtze.hku.hk/lecture/chem3504-3.pptContents

1.VariationMethod2.Hartree-FockSelf-ConsistentFieldMethod3.PerturbationTheory4.SemiempiricalMethodsTheVariationMethodConsiderasystemwhoseHamiltonianoperatorHistimeindependentandwhoselowest-energyeigenvalueisE1.Iffisanynormalized,well-behavedfunctionthatsatisfiestheboundaryconditionsoftheproblem,then

f*H

fdt>

E1ThevariationtheoremProof:Expandfinthebasisset{yk}

f=

k

akykwhere

{ak}arecoefficients

Hyk=Ekykthen

f*H

fdt=

k

j

ak*ajEjdkj

=

k|ak|2

Ek

>E1

k|ak|2=E1

Sinceisnormalized,

f*fdt=

k|ak|2=1

i.f:trialfunctionisusedtoevaluatetheupperlimit

ofgroundstateenergyE1ii.f

=groundstatewavefunction,

f*H

fdt=E1iii.optimizeparamemtersinfbyminimizing

f*H

fdt/

f*fdt

Requirementsforthetrialwavefunction:i.zeroatboundary;ii.smoothness

amaximuminthecenter.

Trialwavefunction:f=x(l-x)Applicationtoaparticleinaboxofinfinitedepth0l

*H

dx=-(h2/8

2m)

(lx-x2)d2(lx-x2)/dx2dx =h2/(4

2m)

(x2-lx)

dx =h2l3/(24

2m)

*

dx=

x2(l-x)2dx=l5/30 E

=5h2/(4

2l2m)

h2/(8ml2)=E1(1)Constructawavefunction

(c1,c2,

,cm)(2)Calculatetheenergyof

:

E

E

(c1,c2,

,cm)(3)Choose{cj*}(i=1,2,

,m)sothatE

isminimum

VariationalMethodExample:one-dimensionalharmonicoscillator

Potential:V(x)=(1/2)kx2=(1/2)m

2x2=2

2m

2x2Trialwavefunctionforthegroundstate:

(x)=exp(-cx2)

*H

dx=-(h2/8

2m)

exp(-cx2)d2[exp(-cx2)]/dx2dx+2

2m

2

x2exp(-2cx2)dx=(h2/4

2m)(

c/8)1/2+

2m

2(

/8c3)1/2

*

dx=

exp(-2cx2)dx=(

/2)1/2c-1/2 E

=W=(h2/8

2m)c+(

2/2)m

2/cTominimizeW, 0=dW/dc=h2/8

2m-(

2/2)m

2c-2 c=2

2

m/h W

=(1/2)h

. . .

E3

y3 E2

y2 E1

y1ExtensionofVariationMethodForawavefunctionfwhichisorthogonaltothegroundstatewavefunctiony1,i.e.

dt

f*y1=0Ef=

dt

f*Hf/

dt

f*f

>

E2

thefirstexcitedstateenergyThetrialwavefunctionf:

dt

f*y1=0

f=

k=1ak

yk

dt

f*y1=|a1|2=0

Ef=

dt

f*Hf/

dt

f*f=

k=2|ak|2Ek/

k=2|ak|2

>

k=2|ak|2E2/

k=2|ak|2=E2

e

++

y1y2f=c1y1+c2y2W=

f*Hfdt/

f*fdt

=(c12

H11+2c1c2

H12

+c22

H22)/(c12+2c1c2

S

+c22)

W(c12+2c1c2

S

+c22)=c12

H11+2c1c2

H12

+c22

H22ApplicationtoH2+Partialderivativewithrespecttoc1

(

W/

c1=0):

W(c1+Sc2)=c1H11+c2H12

Partialderivativewithrespecttoc2

(

W/

c2=0):W(Sc1+c2)=c1H12+c2H22

(H11-W)c1+(H12-SW)c2=0(H12-SW)c1+(H22-

W)c2=0Tohavenontrivialsolution:

H11-W H12-SW H12-SW H22-

W

ForH2+,

H11=H22;H12<0.

GroundState:Eg=W1=(H11+H12)/(1+S)

f1

=(y1+y2)/

2(1+S)1/2

ExcitedState:Ee=W2=(H11-H12)/(1-S)

f2

=(y1-y2)/

2(1-S)1/2

=0bondingorbitalAnti-bondingorbital

Results:De=1.76eV,Re=1.32A

Exact:De=2.79eV,Re=1.06A

1eV=23.0605kcal/molTrialwavefunction:k3/2

p-1/2

exp(-kr)

Eg=W1(k,R)

ateachR,choosek

sothat

W1/

k=0Results: De=2.36eV,Re=1.06A

Resutls:De=2.73eV,Re=1.06A1s2pInclusionofotheratomicorbitalsFurtherImprovements

H p-1/2

exp(-r)He+ 23/2

p-1/2

exp(-2r)Optimizationof1sorbitals

a11x1+a12x2=b1a21x1+a22x2=b2

(a11a22-a12a21)x1=b1a22-b2a12(a11a22-a12a21)x2=b2a11-b1a21LinearEquations1.twolinearequationsfortwounknown,x1andx2Introducingdeterminant:

a11 a12

=a11a22-a12a21 a21 a22

a11 a12

b1 a12

x1= a21 a22

b2 a22

a11 a12

a11 b1

x2=

a21 a22

a21 b2

Ourcase:b1=b2=0,homogeneous

1.trivialsolution:x1=x2=0

2.nontrivialsolution:

a11 a12

=0 a21 a22

nlinearequationsfornunknownvariablesa11x1+a12x2+...+a1nxn=b1a21x1+a22x2+...+a2nxn=b2an1x1+an2x2+...+annxn=bn

a11 a12 ...a1,k-1b1a1,k+1 ...a1n a21 a22 ...a2,k-1b2a2,k+1 ...a2ndet(aij)xk=. . .... .. .... an1 an2 ...an,k-1b2an,k+1 ...ann

where, a11 a12 ... a1n

a21 a22 ... a2n

det(aij)= . . ... . an1 an2 ... ann

a11 a12 ... a1,k-1 b1 a1,k+1 ... a1n a21 a22 ... a2,k-1 b2 a2,k+1 ... a2n . . ... . . . ... . an1 an2 ... an,k-1 b2 an,k+1 ... annxk= det(aij)

inhomogeneouscase:bk=0foratleastonek(a)travialcase:xk=0,k=1,2,...,n(b)nontravialcase:det(aij)=0

homogeneouscase:bk=0,k=1,2,...,nForan-thorderdeterminant, ndet(aij)=

alkClkl=1

where,ClkiscalledcofactorTrialwavefunctionfisavariationfunctionwhichisacombinationofnlinearindependentfunctions{f1,f2,...fn},

f=c1f1+c2f2+...+cnfn

n

[(Hik-SikW)ck]=0i=1,2,...,n

k=1 Sik

dt

fifk Hik

dt

fiHfk W

dtfH

f/

dtff

(i)W1

W2

...

Wn

arenrootsofEq.(1),(ii)E1

E2

...

En

En+1

...areenergies

ofeigenstates;

then,W1

E1,W2

E2,...,Wn

EnLinearvariationaltheoremMolecularOrbital(MO):

=c1

1+c2

2

(H11-W)c1+(H12-SW)c2=0

S11=1

(H21-SW)c1+(H22-W)c2=0

S22=1Generally:

i

asetofatomicorbitals,basissetLCAO-MO

=c1

1+c2

2++cn

nlinearcombinationofatomicorbitalsn

(Hik-SikW)ck=0i=1,2,,nk=1Hik

dt

i*H

k

Sik

dt

i*

k Skk=1Hamiltonian

H=

(-h2/2ma)

2-(h2/2me)

i

i2

+

ZaZbe2/rab-

i

Zae2/ria

+

i

j

e2/rij

Hy(ri;ra)=Ey(ri;ra)TheBorn-OppenheimerApproximation(1)y(ri;ra)=yel(ri;ra)yN(ra)(2)Hel(ra)=-(h2/2me)

i

i2

-

i

Zae2/ria

+

i

j

e2/rij

VNN=

bZaZbe2/rabHel(ra)

yel(ri;ra)=Eel(ra)

yel(ri;ra)(3)HN=

(-h2/2ma)

2+

U(ra)U(ra)=Eel(ra)+VNNHN(ra)

yN(ra)=EyN(ra)TheBorn-OppenheimerApproximation:AssignmentCalculatethegroundstateenergyandbondlengthofH2usingtheHyperChemwiththe6-31G(Hint:Born-OppenheimerApproximation)

e

++

etwoelectronscannotbeinthesamestate.HydrogenMoleculeH2ThePauliprincipleSincetwowavefunctionsthatcorrespondtothesamestatecandifferatmostbyaconstantfactor

f(1,2)=c2

f(2,1)

ja(1)jb(2)+c1ja(2)jb(1)=c2ja(2)jb(1)+c2c1ja(1)jb(2)c1=c2 c2c1=1Therefore:

c1=c2=

1AccordingtothePauliprinciple, c1=c2=-1Wavefunction:f(1,2)=ja(1)jb(2)+c1ja(2)jb(1)f(2,1)=ja(2)jb(1)+c1ja(1)jb(2)

WavefunctionfofH2:

y(1,2)=1/

2![f(1)a(1)f(2)b(2)-f(2)a(2)f(1)b(1)]f(1)a(1)f(2)a(2)=1/

2!

f(1)b(1)f(2)b(2)ThePauliprinciple(differentversion)thewavefunctionofasystemofelectronsmustbeantisymmetricwithrespecttointerchangingofanytwoelectrons.SlaterDeterminantE

=2

dt1f*(1)(Te+VeN)f(1)+VNN

+

dt1dt2|f2(1)|e2/r12|f2(2)|=

i=1,2

fii+J12+VNN

TominimizeE

undertheconstraint

dt|f2|

=1,use

Lagrange’smethod:

L=E

-2e[

dt1|f2(1)|

-1]dL=dE

-4e

dt1f*(1)df(1)

=4

dt1df*(1)(Te+VeN)f(1)+4

dt1dt2f*(1)f*(2)e2/r12f(2)df(1) -4e

dt1f*(1)df(1)=0

Energy:E

[Te+VeN+

dt2f*(2)e2/r12f(2)]f(1)=ef(1)

(f+J)f=eff(1)=Te(1)+VeN(1) oneelectronoperatorJ(1)=

dt2f*(2)e2/r12f(2)twoelectronCoulomboperator

AverageHamiltonianHartree-Fockequationf(1)istheHamiltonianofelectron1intheabsenceofelectron2;J(1)isthemeanCoulombrepulsionexertedonelectron1by2;e

istheenergyoforbitalf.LCAO-MO: f=c1y1+c2y2

Multipley1fromtheleftandthenintegrate:c1F11+c2F12=e(c1+Sc2)Multipley2

fromtheleftandthenintegrate:

c1F12+c2F22=e(Sc1+c2)

where,

Fij=

dtyi*

(f+J

)yj=Hij+

dtyi*

J

yjS=

dty1

y2 (F11-e)c1+(F12-Se)c2=0 (F12-Se)c1+(F22-

e)c2=0SecularEquation:

F11-e

F12-Se

=0

F12-Se

F22-

e

bondingorbital: e1=(F11+F12)/(1+S)

f1

=(y1+y2)/

2(1+S)1/2

antibondingorbital: e2=(F11-F12)/(1-S)

f2

=(y1-y2)/

2(1-S)1/2MolecularOrbitalConfigurationsofHomonuclearDiatomicMoleculesH2,Li2,O,He2,etcMoeculeBondorderDe/eVH2+

2.79

H214.75He2+1.08He200.0009Li211.07Be200.10C226.3N2+

8.85N239.91O2+26.78O225.21ThemoretheBondOrderis,thestrongerthechemicalbondis.BondOrder:one-halfthedifferencebetweenthenumberofbondingandantibondingelectrons

f1

f2

f1(1)a(1)f2(1)a(1)y(1,2)=1/

2

f1(2)a(2)f2(2)a(2)=1/

2[f1(1)f2(2)-f2(1)

f1(2)]a(1)a(2)

Ey=

dt1dt2y*Hy=

dt1dt2y*(T1+V1N+T2+V2N+V12+VNN)y=<f1(1)|T1+V1N|f1(1)>+<f2(2)|T2+V2N|f2(2)>+<f1(1)f2(2)|V12|f1(1)f2(2)>-<f1(2)f2(1)|V12|f1(1)f2(2)>+VNN=

i

<fi(1)|T1+V1N|fi(1)>

+<f1(1)f2(2)|V12|f1(1)f2(2)>-<f1(2)f2(1)|V12|f1(1)f2(2)>+

VNN=

i=1,2

fii+J12-

K12+VNNParticleOne:

f(1)+J2(1)

-

K2(1)ParticleTwo: f(2)+J1(2)

-

K1(2)

f(j)

-(h2/2me)

j2-

Za/rja

Jj(1)q(1)

q(1)

dr2fj*(2)e2/r12fj(2)

Kj(1)q(1)

fj(1)

dr2fj*(2)e2/r12q(2)AverageHamiltonian[f(1)+J2(1)-

K2(1)]f1(1)=e1f1(1)[f(2)+J1(2)-

K1(2)]f2(2)=e2f2(2)F(1)

f(1)+J2(1)-

K2(1) Fockoperatorfor1F(2)

f(2)+J1(2)-

K1(2) Fockoperatorfor2

Hartree-FockEquation:FockOperator:1. AttheHartree-FockLeveltherearetwopossible Coulombintegralscontributingtheenergybetween twoelectronsiandj:CoulombintegralsJijand exchangeintegralKij;

2. Fortwoelectronswithdifferentspins,thereisonly CoulombintegralJij;3.Fortwoelectronswiththesamespins,both Coulombandexchangeintegralsexist. Summary4. TotalHartree-Fockenergyconsistsofthe contributionsfromone-electronintegralsfiiand two-electronCoulombintegralsJijandexchange integralsKij;

5. AttheHartree-FockLeveltherearetwopossible Coulombpotentials(oroperators)betweentwo electronsiandj:Coulomboperatorandexchange operator;Jj(i)istheCoulombpotential(operator) thatifeelsfromj,andKj(i)istheexchange potential(operator)thatthatifeelsfromj.

6.Fockoperator(or,averageHamiltonian)consistsofone-electronoperatorsf(i)andCoulomboperatorsJj(i)

andexchangeoperatorsKj(i)

Na

electronsspinupandNb

electronsspindown.

Fockmatrixforanelectron1

withspinup:

Fa(1

)=fa(1

)+

j[Jja(1

)-Kja(1

)]+

j

Jjb(1

)

j=1

,Na

j=1

,Nb

Fockmatrixforanelectron1

withspindown:

Fb(1

)=fb(1

)+

j[Jjb(1

)-Kjb(1

)]+

j

Jja(1

) j=1

,Nb

j=1

,Na

f(1)

-(h2/2me)

12-

NZN/r1N

Jja(1)

dr2fja*(2)

e2/r12fja(2)Kja(1)q(1)

fja(1)

dr2fja*(2)e2/r12q(2)

Energy=

ja

fjja+

jb

fjjb+(1/2)

ia

ja(Jijaa

-Kijaa)+(1/2)

ia

jb(Jijbb

-Kijbb)+

ia

jb

Jijab

+VNNi=1,Na

j=1,Nbfjj

fjja

<fja|f|fja>Jij

Jijaa

<faj(2)|Jia(1)

|faj(2)>Kij

Kijaa

<faj(2)|Kia(1)

|faj(2)>Jij

Jijab

<fbj(2)|Jia(1)

|fbj(2)>

F(1)=f(1)+

j=1,n/2[2Jj(1)-Kj(1)]

Energy=2

j=1,n/2

fjj+

i=1,n/2

j=1,n/2(2Jij

-Kij)+VNNClosesubshellcase:(Na=Nb=n/2)1.Many-BodyWaveFunctionisapproximated bySlaterDeterminant2.Hartree-FockEquation F

fi=eifi

F

Fockoperator fi thei-thHartree-Fockorbital ei theenergyofthei-thHartree-FockorbitalHartree-FockMethod3.RoothaanMethod(introductionofBasisfunctions)fi

=

kcki

yk LCAO-MO

{yk}

isasetofatomicorbitals(orbasisfunctions)4.Hartree-Fock-Roothaan

equation

j(Fij-eiSij)cji=0

Fij

<

i|F|

j> Sij

<

i|

j>5.SolvetheHartree-Fock-Roothaanequation

self-consistently

<fa(1)fb(2)fc(3)...fd(n)|f(1)|fe(1)ff(2)fg(3)...fh(n)>=<fa(1)|f(1)|fe(1)><fb(2)fc(3)...fd(n)|ff(2)fg(3)...fh(n)>=<fa(1)|f(1)|fe(1)>

if

b=f,c=g,...,d=h;0,otherwise

<fa(1)fb(2)fc(3)...fd(n)|V12|fe(1)ff(2)fg(3)...fh(n)>=<fa(1)fb(2)|V12|fe(1)ff(2)><fc(3)...fd(n)|fg(3)...fh(n)>=<fa(1)fb(2)|V12|fe(1)ff(2)>

if

c=g,...,d=h;0,otherwiseTheCondon-SlaterRules

thelowestunoccupiedmolecularorbital

thehighestoccupiedmolecularorbital

Theenergyrequiredtoremoveanelectronfromaclosed-shellatomormoleculesiswellapproximatedbyminustheorbitalenergyeoftheAOorMOfromwhichtheelectronisremoved.HOMOLUMOKoopman’sTheorem#HF/6-31G(d)Routesection

waterenergyTitle01MoleculeSpecificationO-0.4640.1770.0(inCartesiancoordinatesH-0.4641.1370.0H0.441-0.1430.0Slater-typeorbitals(STO)

nlm=N

rn-1exp(-

r/a0)Ylm(

,

) x

theorbital

exponent*

isusedinsteadof

inthetextbookGaussiantypefunctionsgijk=Nxiyjzkexp(-ar2)(primitiveGaussianfunction)

p=

udup

gu(contractedGaussian-typefunction,CGTF)u={ijk} p={nlm}BasisSet

i=

pcip

pBasissetofGTFs

STO-3G,3-21G,4-31G,6-31G,6-31G*,6-31G**

complexity&accuracyMinimalbasisset:oneSTOforeachatomicorbital(AO)STO-3G:3GTFsforeachatomicorbital3-21G:3GTFsforeachinnershellAO2CGTFs(w/2&1GTFs)foreachvalenceAO6-31G:6GTFsforeachinnershellAO2CGTFs(w/3&1GTFs)foreachvalenceAO6-31G*:addsasetofdorbitalstoatomsin2nd&3rdrows6-31G**:addsasetofdorbitalstoatomsin2nd&3rdrows andasetofpfunctionstohydrogenPolarizationFunctionDiffuseBasisSets:Forexcitedstatesandinanionswhereelectronicdensityismorespreadout,additionalbasisfunctionsareneeded.Diffusefunctionsto6-31Gbasissetasfollows:

6-31G*-addsasetofdiffuses&porbitalstoatomsin1st&2ndrows(Li-Cl).6-31G**-addsasetofdiffusesandporbitalstoatomsin1st&2ndrows(Li-Cl)andasetofdiffusesfunctionstoH

Diffusefunctions+polarisationfunctions:6-31+G*,6-31++G*,6-31+G**and6-31++G**basissets.Double-zeta(DZ)basisset:

twoSTOforeachAO6-31Gforacarbonatom: (10s4p)

[3s2p] 1s 2s 2pi(i=x,y,z)

6GTFs 3GTFs 1GTF 3GTFs1GTF1CGTF1CGTF1CGTF1CGTF1CGTF (s) (s) (s)(p)(p)Minimalbasisset:

OneSTOforeachinner-shellandvalence-shellAOofeachatom

example:C2H2(2S1P/1S)C:1S,2S,2Px,2Py,2PzH:1Stotal12STOsasBasissetDouble-Zeta(DZ)basisset:twoSTOsforeachandvalence-shellAOofeachatomexample:C2H2(4S2P/2S)C:two1S,two2S,two2Px,two2Py,two2PzH:two1S(STOs)total24STOsasBasissetSplit-Valence(SV)basissetTwoSTOsforeachinner-shellandvalence-shellAOOneSTOforeachinner-shellAODouble-zetapluspolarizationset(DZ+P,orDZP)AdditionalSTOw/lquantumnumberlargerthanthelmaxofthevalence-shell(2Px,2Py,2Pz)toHFive3dAostoLi-Ne,Na-ArC2H5OSiH3:(6s4p1d/4s2p1d/2s1p)SiC,OHAssignment:Calculatethestructure,groundstateenergy,molecularorbitalenergies,andvibrationalmodesandfrequenciesofawatermoleculeusingHartree-Fockmethodwith3-21Gbasisset.(due30/10)1.L-Clickon(click