11.2: Gaussian Basis Sets - Chemistry LibreTexts

Gaussian basis sets are identified by abbreviations such as N-MPG*. N is the number of Gaussian primitives used for each inner-shell orbital. Skiptomaincontent TheVariationalMethodandBasisSetsSlaterTypeOrbitals(STOs)Double-zetabasisSetsGaussianOrbitalsSummaryContributors Abasissetintheoreticalandcomputationalchemistryisasetoffunctions(calledbasisfunctions)whicharecombinedinlinearcombinations(generallyaspartofaquantumchemicalcalculation)tocreatemolecularorbitals.Forconveniencethesefunctionsaretypicallyatomicorbitalscenteredonatoms,butcantheoreticallybeanyfunction;planewavesarefrequentlyusedinmaterialscalculations. TheVariationalMethodandBasisSets Todescribetheelectronicstatesofmolecules,weconstructwavefunctionsfortheelectronicstatesbyusingmolecularorbitals.ThesewavefunctionsareapproximatesolutionstotheSchrödingerequation.Amathematicalfunctionforamolecularorbitalisconstructed,\(\psi_i\),asalinearcombinationofotherfunctions,\(\varphi_j\),whicharecalledbasisfunctionsbecausetheyprovidethebasisforrepresentingthemolecularorbital. \[\psi_i=\sum_jc_{ij}\varphi_j\label{10.8}\] Thevariationalmethodisusedtofindvaluesforparametersinthebasisfunctionsandfortheconstantcoefficientsinthelinearcombinationthatoptimizethesefunctions,i.e.makethemasgoodaspossible.Thecriterionforqualityinthevariationalmethodismakingthegroundstateenergyofthemoleculeaslowaspossible.Hereandintherestofthischapter,thefollowingnotationisused:\(\sigma\)isageneralspinfunction(canbeeither\(\alpha\)or\(\beta\)),\(\varphi\)isthebasisfunction(thisusuallyrepresentsanatomicorbital),\(\psi\)isamolecularorbital,and\(\Psi\)istheelectronicstatewavefunction(representingasingleSlaterdeterminantorlinearcombinationofSlaterdeterminants). Theultimategoalisamathematicaldescriptionofelectronsinmoleculesthatenableschemistsandotherscientiststodevelopadeepunderstandingofchemicalbondingandreactivity,tocalculatepropertiesofmolecules,andtomakepredictionsbasedonthesecalculations.Forexample,anactiveareaofresearchinindustryinvolvescalculatingchangesinchemicalpropertiesofpharmaceuticaldrugsasaresultofchangesinchemicalstructure. Selectingtheabinitiomodelforachemicalsystemisalmostalwaysinvolvesatrade-offbetweenaccuracyandcomputationalcost.Moreaccuratemethodsandlargerbasissetsmakejobsrunlonger. Inmoderncomputationalchemistry,quantumchemicalcalculationsaretypicallyperformedusingafinitesetofbasisfunctions.Inthesecases,thewavefunctionsofthesysteminquestionarerepresentedasvectors,thecomponentsofwhichcorrespondtocoefficientsinalinearcombinationofthebasisfunctionsinthebasissetused. Themolecularspin-orbitalsthatareusedintheSlaterdeterminantusuallyareexpressedasalinearcombinationofsomechosenfunctions,whicharecalledbasisfunctions.Thissetoffunctionsiscalledthebasisset.Thefactthatonefunctioncanberepresentedbyalinearcombinationofotherfunctionsisageneralproperty.Allthatisnecessaryisthatthebasisfunctionsspan-the-space,whichmeansthatthefunctionsmustformacompletesetandmustbedescribingthesamething.Forexample,sphericalharmonicscannotbeusedtodescribeahydrogenatomradialfunctionbecausetheydonotinvolvethedistancer,buttheycanbeusedtodescribetheangularpropertiesofanythinginthree-dimensionalspace. Thisspan-the-spacepropertyoffunctionsisjustlikethecorrespondingpropertyofvectors.Theunitvectors\((\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})\)describepointsinspaceandformacompletesetsinceanypositioninspacecanbespecifiedbyalinearcombinationofthesethreeunitvectors.Theseunitvectorsalsocouldbecalledbasisvectors. Exercise\(\PageIndex{1}\):"SpanningtheSpace" Explainwhytheunitvectors\((\overrightarrow{x},\overrightarrow{y})\)donotformacompletesettodescribeyour(three-dimensional)classroom. Justaswediscussedforatoms,parametersinthebasisfunctionsandthecoefficientsinthelinearcombinationcanbeoptimizedinaccordwiththeVariationalPrincipletoproduceaself-consistentfield(SCF)fortheelectrons.Thisoptimizationmeansthatthegroundstateenergycalculatedwiththewavefunctionisminimizedwithrespecttovariationoftheparametersandcoefficientsdefiningthefunction.Asaresult,thatgroundstateenergyislargerthantheexactenergy,butisthebestvaluethatcanbeobtainedwiththatwavefunction. SlaterTypeOrbitals(STOs) Intuitivelyonemightselecthydrogenicatomicorbitalsasthebasissetformolecularorbitals.Afterall,moleculesarecomposedofatoms,andhydrogenicorbitalsdescribeatomsexactlyiftheelectron-electroninteractionsareneglected.Atabetterlevelofapproximation,thenuclearchargethatappearsinthesefunctionscanbeusedasavariationalparametertoaccountfortheshieldingeffectsduetotheelectron-electroninteractions.Also,theuseofatomicorbitalsallowsustointerpretmolecularpropertiesandchargedistributionsintermsofatomicpropertiesandcharges,whichisveryappealingsincewepicturemoleculesascomposedofatoms.Asdescribedinthepreviouschapter,calculationswithhydrogenicfunctionswerenotveryefficientsootherbasisfunctions,Slater-typeatomicorbitals(STOs),wereinvented. AminimalbasissetofSTOsforamoleculeincludesonlythoseSTOsthatwouldbeoccupiedbyelectronsintheatomsformingthemolecule.Alargerbasisset,however,improvestheaccuracyofthecalculationsbyprovidingmorevariableparameterstoproduceabetterapproximatewavefunction,butattheexpenseofincreasedcomputationaltime.STOshavethefollowingradialpart(thesphericalharmonicfunctionsareusedtodescribetheangularpart) \[R(r)=Nr^{n−1}e^{−\zetar}\] where \(n\)isanaturalnumberthatplaystheroleofprincipalquantumnumber,n=1,2,..., \(N\)isanormalizingconstant, \(r\)isthedistanceoftheelectronfromtheatomicnucleus,and\(\zeta\)isaconstantrelatedtotheeffectivechargeofthenucleus,thenuclearchargebeingpartlyshieldedbyelectrons.Historically,theeffectivenuclearchargewasestimatedbySlater'srules. Double-zetabasisSets OnecanusemorethanoneSTOtorepresentoneatomicorbital,asshowninEquation\(\ref{10.11}\),andratherthandoinganonlinearvariationalcalculationtooptimizeeach\(\zeta\)value,usetwoSTOswithdifferent\(\zeta\)variables.Thelinearvariationcalculationthenwillproducethecoefficients(\(C_1\)and\(C_2\))forthesetwofunctionsinthelinearcombinationthatbestdescribesthechargedistributioninthemolecule(forthegroundstate).Thefunctionwiththelargezetaaccountsforchargenearthenucleus,whilethefunctionwiththesmallerzetaaccountsforthechargedistributionatlargervaluesofthedistancefromthenucleus.Thisexpandedbasissetiscalledadouble-zetabasisset. \[R_{2s}(r)=C_1re^{-\zeta_1r}+C_2re^{-\zeta_2r}\label{10.11}\] Theuseofdoublezetafunctionsinbasissetsisespeciallyimportantbecausewithoutthemorbitalsofthesametypeareconstrainedtobeidenticaleventhoughinthemoleculetheymaybechemicallyinequivalent.Forexample,inacetylenethe\(p_z\)orbitalalongtheinternuclearaxisisinaquitedifferentchemicalenvironmentandisbeingusedtoaccountforquitedifferentbondingthanthe\(p_x\)and\(p_y\)orbitals.Withadoublezetabasissetthe\(p_z\)orbitalisnotconstrainedtobethesamesizeasthe\(p_x\)and\(p_y\)orbitals. Example\(\PageIndex{1}\) Explainwhythe\(p_x\),\(p_y\),and\(p_z\)orbitalsinamoleculemightbeconstrainedtobethesameinasingle-zetabasissetcalculation,andhowtheuseofadouble-zetabasissetwouldallowthe\(p_x\),\(p_y\),and\(p_z\)orbitalstodiffer. GaussianOrbitals Althoughanybasissetthatsufficientlyspansthespaceofelectrondistributioncouldbeused,theconceptofMolecularOrbitalsasLinearCombinationsofAtomicOrbitals(LCAO)suggestsaverynaturalsetofbasisfunctions:AO-typefunctionscenteredoneachnuclei.OneobviouschoicearetheexacthydrogenAO's,knownasSlater-typeorbitals(STO)--describingtheradialcomponentofthefunctions.However,thecomputationoftheintegralsisgreatlysimplifiedbyusingGaussian-typeorbitals(GTO)forbasisfunctions. WhiletheSTObasissetwasanimprovementoverhydrogenicorbitalsintermsofcomputationalefficiency,representingtheSTOswithGaussianfunctionsproducedfurtherimprovementsthatwereneededtoaccuratelydescribemolecules.AGaussianbasisfunctionhastheformshowninEquation\(\ref{10.12}\).Notethatinallthebasissets,onlytheradialpartoftheorbitalchanges,andthesphericalharmonicfunctionsareusedinallofthemtodescribetheangularpartoftheorbital. \[G_{nlm}(r,\theta,\psi)=N_n\underbrace{r^{n-1}e^{-\alphar^2}}_{\text{radialpart}}\underbrace{Y^m_l(\theta,\psi)}_{\text{angularpart}}\label{10.12}\] UnfortunatelyGaussianfunctionsdonotmatchtheshapeofanatomicorbitalverywell.Inparticular,theyareflatratherthansteepneartheatomicnucleusat\(r=0\),andtheyfalloffmorerapidlyatlargevaluesof\(r\)(Figure\(\PageIndex{1}\)). Figure\(\PageIndex{1}\):RadialDependenceofSlaterandGaussianBasisFunctions.Imageusedwithpermission. Tocompensateforthisproblem,eachSTOisreplacedwithanumberofGaussianfunctionswithdifferentvaluesfortheexponentialparameter.TheseGaussianfunctionsformaprimitiveGaussianbasisset.LinearcombinationsoftheprimitiveGaussiansareformedtoapproximatetheradialpartofanSTO.Thislinearcombinationisnotoptimizedfurtherintheenergyvariationalcalculation,butratherisfrozenandtreatedasasinglefunction.ThelinearcombinationofprimitiveGaussianfunctionsiscalledacontractedGaussianfunction.Althoughmorefunctionsandmoreintegralsnowarepartofthecalculation,theintegralsinvolvingGaussianfunctionsarequickertocomputethanthoseinvolvingexponentials,sothereisanetgainintheefficiencyofthecalculation. Figure\(\PageIndex{2}\):Tobetterrepresentthecuspintheelectrondensityatthenuclei,GTObasissetsareconstructedfromfixedlinear-combinationsofGaussianfunctions,contractedGTOs(CGTO).TheearliestCGTObasissets,whereconstructedfromNGTOsthatbestfitthedesiredSTO.ThesearecalledSTO-NGbasissets. GaussianbasissetsareidentifiedbyabbreviationssuchasN-MPG*.NisthenumberofGaussianprimitivesusedforeachinner-shellorbital.Thehyphenindicatesasplit-basissetwherethevalenceorbitalsaredoublezeta.TheMindicatesthenumberofprimitivesthatformthelargezetafunction(fortheinnervalenceregion),andPindicatesthenumberthatformthesmallzetafunction(fortheoutervalenceregion).GidentifiesthesetabeingGaussian.TheadditionofanasterisktothisnotationmeansthatasinglesetofGaussian3dpolarizationfunctions(discussedelswhere)isincluded.AdoubleasteriskmeansthatasinglesetofGaussian2pfunctionsisincludedforeachhydrogenatom. Forexample,3GmeanseachSTOisrepresentedbyalinearcombinationofthreeprimitiveGaussianfunctions.6-31Gmeanseachinnershell(1sorbital)STOisalinearcombinationof6primitivesandeachvalenceshellSTOissplitintoaninnerandouterpart(doublezeta)using3and1primitiveGaussians,respectively(seeTable\(\PageIndex{1}\)forotherexamples). Basisset #functions Basisset #functions Basisset #functions Table\(\PageIndex{1}\):DifferentGaussianBasissets STO-3G 5 6-31G 9 6-311G 13 3-21G 9 6-31G* 15 6-311G* 18* 4-31G 9 6-31+G* 19 6-311+G* 22* Example\(\PageIndex{2}\) The1sSlater-typeorbital\(S_1(r)=\sqrt{4\zeta_1e^{-\zeta_1r}}\)with\(\zeta_1=1.24\)isrepresentedasasumofthreeprimitiveGaussianfunctions, \[S_G(r)=\sum_{j=1}^3C_je^{-\alpha_jr^2}\nonumber\] ThissumisthecontractedGaussianfunctionfortheSTO. MakeplotsoftheSTOandthecontractedGaussianfunctiononthesamegraphsotheycanbecomparedeasily.AlldistancesshouldbeinunitsoftheBohrradius.Usethefollowingvaluesforthecoefficients,C,andtheexponentialparameters,\(\alpha\). indexj \(\alpha_j\) \(C_j\) 1 0.1688 0.4 2 0.6239 0.7 3 3.425 1.3 Changethevaluesofthecoefficientsandexponentialparameterstoseeifabetterfitcanbeobtained. CommentontheabilityofalinearcombinationofGaussianfunctionstoaccuratelydescribeaSTO. Summary Whenmolecularcalculationsareperformed,itiscommontouseabasiscomposedofafinitenumberofatomicorbitals(Equation\(\ref{10.8}\)),centeredateachatomicnucleuswithinthemolecule(linearcombinationofatomicorbitalsansatz).TheseatomicorbitalsarewelldescribedwithSlater-typeorbitals(STOs),asSTOsdecayexponentiallywithdistancefromthenuclei,accuratelydescribingthelong-rangeoverlapbetweenatoms,andreachamaximumatzero,welldescribingthechargeandspinatthenucleus.STOsarecomputationallydifficultanditwaslaterrealizedbyFrankBoysthattheseSlater-typeorbitalscouldinturnbeapproximatedaslinearcombinationsofGaussianorbitalsinstead.BecauseitiseasiertocalculateoverlapandotherintegralswithGaussianbasisfunctions,thisledtohugecomputationalsavings Contributors Wikipedia DavidM.Hanson,EricaHarvey,RobertSweeney,TheresaJuliaZielinski("QuantumStatesofAtomsandMolecules")