Lecture 8 Gaussian basis sets - University of Southampton

1 CHEM6085 Density Functional Theory1 Lecture 8 Gaussian basis SkylarisCHEM6085: Density Functional Theory2 WheretheKohn-Shampotentialofthenon-inter actingelectronsisgivenby TheSCFprocedureinvolvessolvingtheKohn-Sh amsingle-electronequationsforthemolecula rorbitals Weallhavesomeexperienceinsolvingequation sonpaperbuthowwedothiswithacomputer?CHEM 6085 Density Functional TheorySolving the Kohn-Sham equations on a computerCHEM6085 Density Functional Theory3 linear combination of atomic orbitals (LCAO) WewillexpresstheMOsasalinearcombinationo fatomicorbitals(LCAO) ThestrengthoftheLCAO approachisitsgeneralapplicability:itcanw orkonanymoleculewithanynumberofatomsACBE xample:AOs on atom AAOs on atom BAOs on atom CCHEM6085 Density Functional Theory4 Example: find the AOs from which the MOs of the following molecules will be built5 basis functions WecantaketheLCAO conceptonestepfurther: UsealargernumberofAOs( ,andsomepanddAOs,etc.)

2 Thiswillachieveamoreflexiblerepresentati onofourMOsandthereforemoreaccuratecalcul atedpropertiesaccordingtothevariationpri nciple UseAOsofaparticularmathematicalformthats implifiesthecomputations(butarenotnecess arilyequaltotheexactAOsoftheisolatedatom s) WecallsuchsetsofmoregeneralAOsbasisfunct ions Insteadofhavingtocalculatethemathematica lformoftheMOs(impossibleonacomputer)thep roblemisreducedtodeterminingtheMOexpansi oncoefficientsintermsofthebasisfunctions molecular (spatial) orbitalbasis function (fixed)coefficient(a number, to be determinedby the SCF calculation)CHEM6085 Density Functional Theory6 Gaussian basis functionsFunctionsthatresemblehydrogenAO s(Slaterfunctions)areverysuitableforexpa ndingMOsbecausetheyhavethecorrectshape: Nearthenucleus(cusp) Farfromthenucleus(decaylikee-ar)However, Gaussianfunctionsarepreferredinpracticeb ecausetheyallowforefficientcomputationof molecularintegrals(simplerformulas)Gauss ian functionSlater functionCHEM6085 Density Functional TheoryCHEM6085 Density Functional Theory7 Primitive Gaussian functions InthejargonofquantumchemistryasingleGaus sianfunctioniscalledaprimitiveGaussianfu nction,orprimitiveGTO(GaussianTypeOrbita l) SomeprogramsuseCartesianprimitiveGTOs.

3 OtherprogramsuseSphericalprimitiveGTOs SphericalandCartesianfunctionsarethesame foruptol=1(pfunctions)butdifferslightlyf orl=2orhigher8 Cartesian primitive Gaussian functionsSimilartoatomicorbitals,wedefin eCartesianGaussianatomicfunctionsbyanang ularmomentum quantumnumber landthenumbersk,mandnwherel=k+m+ functionsp functionsd functionsCHEM6085 Density Functional Theory9 Spherical primitive Gaussian functionsSimilartoatomicorbitals,wedefin eSphericalGaussianatomicfunctionsbyanang ularmomentum quantumnumber landitscomponentsm=-l,.., functionsp functionsd functionsCHEM6085 Density Functional Theory Inpractice,fixedlinearcombinationsof primitive Gaussianfunctionsareused Thesearecalled ContractedGaussians (CGs): ThesimplestkindofCGsaretheSTO-nGbasisset s ThesebasissetsattempttoapproximateSlater -typeorbitals(STOs)bynprimitiveGaussians 10 Contracted Gaussian Density Functional Theory11 Gaussian basis setsTheSTO-nGbasissetsareratherunsatisfa ctoryastheyincludeonlyonecontractedGauss sian(CG) : MorethanoneCGperatomicorbital, :DZ( doublezeta ),TZ( triplezeta ),QZ( quadruplezeta ) OneCGper core atomicorbitalandmorethanoneforthevalence atomicorbitals, :SV,3-21G,4-31G,6-31G,6-311 GWritedownhowmanyCGssomeoftheabovebasiss etswillincludeforthefollowingatoms.

4 H,C,SAndforthefollowingmolecules:H2O,CH4 CHEM6085 Density Functional Theory12 Polarisation and diffuse ,suchas: polarisationfunctions enhancethe flexibility :3-21G*,6-31G*,6-31G**,DVP,TZP,cc-pVDZ,c c-pVTZ diffusefunctions :4-31+G,6-31+G Basissetsareconsidered balanced :6-31+G*,6-311++G**,aug-cc-pVDZW ritedownhowmanyCGssomeoftheabovebasisset swillincludeforthefollowingatoms:H,F,San dmolecules:H2O,CH4 CHEM6085 Density Functional Theory13 The complete basis set limit Basissetsareanapproximationintroducedino rdertosolvetheKSequationsfortheMOsonacom puter TheMOsobtainedaresolutionsoftheKohn-Sham equationsonlywithinthe functionspace ofthebasissetused(sosolutionswithintheST O-3 Gsetoffunctions,orthe6-31 Gset,etc.) Improvingthequalityofthebasissetrequires increasingthenumberofCGs Acompletebasissetcanrepresentexactlyanym olecularorbital Unfortunately,completebasissetstendtohav eaninfinitenumberoffunctionsandaretheref orenotpracticalforcalculationsCHEM6085 Density Functional Theory14 Extrapolation to the complete basis set limit Wecanestimatethecompletebasissetresultby systematicallyincreasingthenumberofbasis functionsandextrapolatingtoaninfinite-si zebasisset Thecc-pVDZ,cc-pVTZ,cc-pVQZ,etc,basissets areanexampleofasystematicseriesofbasisse tsthatcanbeextrapolatedtothecompletebasi ssetlimitCHEM6085 Density Functional TheoryCHEM6085 Density Functional Theory15 Example calculations.

5 Protonationenergy of a water moleculeBasis setH2O energy(Eh)H3O+energy (Eh)Protonationenergy (Eh)Protonationenergy (kcal/mol) ++ ** ++G** +++CHEM6085 Density Functional Theory163-21g aug-cc-pvqzcc-pvtzfi_sf_fwlanl2dzdp_ecp_ polarization3-21++g aug-cc-pvqz_diffusecc-pvtzfi_sf_lclanl2d z_ecp3-21gs aug-cc-pvt+dzcc-pvtzfi_sf_scmclean_chand ler_vtz3-21++gsaug-cc-pvt+dz_diffusecc-p vtz-fit2-1 midi!3-21gsp aug-cc-pvtzcc-pvtzpt_sf_fwmidi_huzinaga3 -21gs_polarization aug-cc-pvtz_diffusecc-pvtzpt_sf_lcmini_h uzinaga4-22gsp bauschlicher_anocc-pvtzpt_sf_scmini_scal ed4-31g binning-curtiss_1d_polarization cc-pvtzseg-opt nasa_ames_ano6-311g binning-curtiss_df_polarizationchipman_d zp_+_diffusenasa_ames_cc-pcv5z6-311++g2d _2p nasa_ames_cc-pcvqz6-311g2df_2pd nasa_ames_cc-pcvtz6-311++g3df_3pd +gs cc-pcv5z cc-pcv6z ++gsscc-pcvdzd-aug-cc-pv5z

6 Cc-pcvqzd-aug-cc-pv5z_diffuse pople_2d_2p_polarization6-31g cc-pcvtzd-aug-cc-pv6z pople_2df_2pd_polarization6-31++g cc-pv5+dz d-aug-cc-pv6z_diffuse pople_3df_3pd_polarization6-31g3df_3pd cc-pv5z d-aug-cc-pvdzpople-style_diffuse6-31g-bl audeau cc-pv5z_dk d-aug-cc-pvdz_diffusepv6z6-31gs cc-pv5zfi_sf_fw d-aug-cc-pvqzqmmm_zhang_3-21g_ecp6-31+gs cc-pv5zfi_sf_lc d-aug-cc-pvqz_diffuseqmmm_zhang_6-31gs_e cp6-31++gscc-pv5zfi_sf_sc d-aug-cc-pvtzsadlej_pvtz6-31gs-blaudeau cc-pv5zpt_sf_fw d-aug-cc-pvtz_diffusesbkjc_vdz_ecp6-31gs _polarization cc-pv5zpt_sf_lc demon_coulomb_fittingsdb-aug-cc-pvqz6-31 gss cc-pv5zpt_sf_sc dgauss_a1_dft_coulomb_fitting sdb-aug-cc-pvqz_diffuse6-31++gsscc-pv6+d z dgauss_a1_dft_exchange_fitting sdb-aug-cc-pvtz6-31gss_polarization cc-pv6z dgauss_a2_dft_coulomb_fitting sdb-aug-cc-pvtz_diffuseahlrichs_coulomb_ fittingcc-pvd+dzdgauss_a2_dft_exchange_f itting

7 Sdb-cc-pvqzahlrichs_polarizationcc-pvdzd hms_polarizationsdb-cc-pvtzahlrichs_pvdz cc-pvdz_dkdunning-hay_diffusesto-2gahlri chs_tzvcc-pvdzfi_sf_fwdunning-hay_double _rydbergsto-3gahlrichs_vdzcc-pvdzfi_sf_l cdunning-hay_rydbergsto-3gsahlrichs_vtzc c-pvdzfi_sf_scdz_+_double_rydberg_dunnin g-hay sto-3gs_polarizationaug-cc-pcv5z cc-pvdz-fit2-1 dz_dunningsto-6gaug-cc-pcvdzcc-pvdzpt_sf _fwdzp_+_diffuse_dunningstuttgart_rlc_ec paug-cc-pcvqzcc-pvdzpt_sf_lcdzp_dunnings tuttgart_rsc_1997_ecpaug-cc-pcvtzcc-pvdz pt_sf_scdzp_+_rydberg_dunningstuttgart_r sc_ano_ecpaug-cc-pv5+dz cc-pvdzseg-opt dz_+_rydberg_dunningstuttgart_rsc_segmen ted_ecpaug-cc-pv5+dz_diffuse cc-pvq+dzdzvp2_dft_orbital sv_+_double_rydberg_dunning-hayaug-cc-pv 5z cc-pvqzdzvp_dft_orbitalsv_dunning-hayaug -cc-pv5z_diffuse +_diffuse_dunning-hayaug-cc-pv6+dz +_diffuse_+_rydbergaug-cc-pv6+dz_diffuse cc-pvqzfi_sf_scgamess_pvtzsvp_+_rydberg_ dunning-hayaug-cc-pv6z_diffuse cc-pvqzpt_sf_fwgamess_vtzsv_+_rydberg_du nning-hayaug-cc-pvd+dzcc-pvqzpt_sf_lcgle ndening_polarizationtz_dunningaug-cc-pvd +dz_diffusecc-pvqzpt_sf_schay-wadt_mb_n+ 1_ecp tzvp_dft_orbitalaug-cc-pvdzcc-pvqzseg-op t hay-wadt_vdz_n+1_ecp wachters+faug-cc-pvdz_diffusecc-pvt+dzho ndo7_polarization

8 Wtbsaug-cc-pvq+dzcc-pvtzhuzinaga_polariz ationaug-cc-pvq+dz_diffusecc-pvtz_dklanl 2dzdp_ecpExample: the available choices of basis set in a quantum chemistry program17 basis sets on the Web Many kinds of basis sets have been developed over the years Most are available for download from websites, such as Density Functional Theory18 Example: STO-3G basis for hydrogenExponents and contraction :1)Exponents2)Contractioncoefficientsexp onentscontraction coefficientsCHEM6085 Density Functional Theory19 Downloading basis setsExample: Data needed for calculation on water with 6-31G** basisCHEM6085 Density Functional Theory1)Describehowmanyandwhattype(s,p,d )primitiveandcontractedGaussiansyouwillh aveintheSTO-2G, )The3-21G*basissetforacarbonatomcanbeinp utintoaquantumchemistryprogramusingthefo llowingparameters(exponentsandcontractio ncoefficients) ,valenceelectronsandforpolarisation?

9 Describehowyoucan uncontract )Substitutetheexpressionforthebasissetex pansionofamolecularorbitalintotheSchr dingerequationfortheKohn-Shamorbitalsand deriveamatrixrepresentationoftheSchr dingerequation,involvingthe matrixelements oftheKohn-ShamHamiltonianandtheoverlapma trixofthebasisfunctions(whicharenotortho gonal).Thismatrixequationcanbesolvedonac omputertoobtaintheorbitalexpansioncoeffi cients(diagonalisationoftheHamiltonianma trix) Density Functional Theory21 Name :Surname:5-minute quiz1)WhyareSlaterfunctionsmoresuitablet hanGaussiansasbasisfunctions?2)Whatdowem eanwhenwesaythatabasissetcontains polarisationfunctions andwhatdowemeanby diffusefunctions ?3)Whatisa double-zetabasisset ?Wouldyouexpectatriple-zetabasissettogiv ebetterresultsthanSTO-3 Gandwhy?

10 Date :CHEM6085 Density Functional Theory