量子化学2

NotesontheHartree-FockMethodXihuaChenDepartmentofChemistry,NewYorkUniversity,NewYork,NY10003,USASpring,2005Contents1ElectronicProblem31.1PostulatesofQuantumMechanics.................31.2Time-independentSchr¨odingerequation..............41.3Atomicunits.............................41.4Born-Oppenheimerapproximation.................61.5Antisymmetryofelectronicwavefunction.............82Hartree-FockApproximation82.1One-electronmolecularorbital...................82.2Slaterdeterminant..........................92.3EnergyoftheHartree-Fockgroundstate..............113Hartree-FockEquations143.1Variationprinciple.........................143.2DerivationofHartree-Fockequations...............163.3ThecanonicalHartree-Fockequations...............213.4OperatorsinHartree-Fockequations................233.5OrbitalenergiesandKoopman’stheorem..............2414RestrictedClose-ShellHartree-Fock264.1Restrictedspinorbitals.......................274.2Groundstateenergywithspatialorbitals..............274.3Hartree-Fockequationswithspatialorbitals............314.4LCAO-MOandtheRoothaanequations..............344.5Matrixformulization........................354.5.1AOmatrixamdMOmatrix..............364.5.2Atomicorbitaloverlapmatrix..............364.5.3Molecularorbitalenergymatrix.............374.5.4Coefficientmatrix....................374.5.5Chargedensitymatrix..................384.5.6Core-Hamiltonianmatrix..............394.5.7Thetwo-electronmatrix.................404.5.8TheFockmatrix.....................424.6Orthogonalizationofbasisfunctions................424.7Self-Consistent-Field(SCF)procedure...............454.8Orbitalenergiesandtheelectronicenergy.............484.9OthermolecularProperties.....................504.10Populationanalysis.........................525UnrestrictedOpen-shellHartree-Fock545.1Unresptrictedspinorbitals.....................545.2UnrestrictedHartree-FockEquations................555.3UHForbitalenergiesandthegroundstateenergy.........565.4BasisfunctionsandPople-NesbetEquations............585.5MatrixformulizationofUHF....................595.6SolvingthePople-Nesbetequations:UHF-SCF..........612ThisdocumentiswrittenasaconciseandcompleteformulizationoftheHartree-Fock(HF)methodtocalculatetheelectronicstructureofatomsandmolecules.Manycomprehensivereviewsandbooksexistonthisfundamentalabinitiomethod.1ElectronicProblem1.1PostulatesofQuantumMechanicsWeshallconsideramolecule(neutralmolecule,radicalorion)whichiscomposedofatomicnucleiandelectrons.Thegeometricalstructureofthemoleculeisdefinedbythepositionsofnuclei,,inaCartesiancoordinatesystem.Nucleiarepositivelychargedwith.Withoutexternalfield,theelectrostaticrepulsionbetweennucleiandbetweenelectrons,andtheelectrostaticattractionbetweennucleiandelectronsthendefinethepotentialofthemolecule.Todescribeamolecularsystemquantummechanically,let’sstartfromthefollowingpostulatesofquantummechanics.Postulate1Atagiventime,,thestateofthemoleculeiscompletelyspecifiedbyawavefunction(orstatefunction),,whichbelongstoastatespaceconsistingofallpossiblestatesofthesystemanddependsonthecoordinatesofnuclei,,andthecoordinatesofelectrons,.Postulate2Foreveryobservable,,inclassicalmechanics,thereexistsacor-respondinglinearoperatorinquantummechanics,.Forexample,theopera-torassociatedwiththetotalenergyofthesystemistheHamiltonianoperator,.Anymeasurementofwillresultinonlyoneoftheeigenval-uesof,.Theaveragevalueoftheobservableisgivenby(1.1)Postulate3Thetimeevolutionofthestatefunctionisgovernedbythetime-dependentSchr¨odingerequation,(1.2)31.2Time-independentSchr¨odingerequationWearemainlyconcernedofthestationarystateofthemolecularsysteminwhichtheHamiltoniandoesn’tcontaintimeexplicitly.Inthiscase,bythemethodofseperationofvariables,wecanwriteEq.(1.2)(1.3)and(1.4)(1.5)Themaininterestofabinitiocalculationissolvingthetime-independentSchr¨odingerequation,(1.3),forthestationarywavefunction.1.3AtomicunitsGivenamolecularsystemofnucleiandelectrons,theHamiltonianassoci-atedwiththetotalenergyisthesumofthekineticoperatorsofnucleiandelectronsandthepotentialoperatorwhichaccountstheelectrostaticrepulsionbetweennu-cleiandbetweenelectrons,andtheelectrostaticattractionbetweennucleiandelectrons.InSIunits,(1.6)whereandarethemassesofanelectronandthethnucleus,respectively;istheunitcharge;isthenumberofpositivecharge4unitsonthethnuclei;,,;isthepermittivityofvacuum;istheLaplacianoperator.Foracertainsystem,andaretheonlyvariables.Inatomicandmolecularcalculations,mostequationswillbegreatlysimplifiedifatomicunits(au)isused.ToderiveaufromSI,wefirstchoose(allto6sig.fig.)masschargeangularmomentumpermittivityThen,let’sconsiderahydrogenatomwiththenucleusfixedattheorigin,andbydefiningaatomicunitoflength,,sothat,casttheaboveequationintoadimensionlessformcanbesuchchosedthattheatomicunitofenergycanbedefinedasthenTherefore,lengthenergyelectricdipolemomentelectricpolarizabilityelectricfieldwavefunction5Inatomicunits,theHamiltonian,Eq.(1.6)canbesimplifiedas(1.7)inwhich.1.4Born-OppenheimerapproximationThelargedifferencebetweenthemassesofthenucleiandtheelectronallowsustonegl