Hybrid functional - Wikipedia

The three parameters defining B3LYP have been taken without modification from Becke's original fitting of the analogous B3PW91 functional to a set of ... Hybridfunctional FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Hybridfunctionalsareaclassofapproximationstotheexchange–correlationenergyfunctionalindensityfunctionaltheory(DFT)thatincorporateaportionofexactexchangefromHartree–Focktheorywiththerestoftheexchange–correlationenergyfromothersources(abinitioorempirical).TheexactexchangeenergyfunctionalisexpressedintermsoftheKohn–Shamorbitalsratherthanthedensity,soistermedanimplicitdensityfunctional.OneofthemostcommonlyusedversionsisB3LYP,whichstandsfor"Becke,3-parameter,Lee–Yang–Parr". Contents 1Origin 2Method 2.1B3LYP 2.2PBE0 2.3HSE 2.4Meta-hybridGGA 3References Origin[edit] ThehybridapproachtoconstructingdensityfunctionalapproximationswasintroducedbyAxelBeckein1993.[1]HybridizationwithHartree–Fock(HF)exchange(alsocalledexactexchange)providesasimpleschemeforimprovingthecalculationofmanymolecularproperties,suchasatomizationenergies,bondlengthsandvibrationfrequencies,whichtendtobepoorlydescribedwithsimple"abinitio"functionals.[2] Method[edit] Ahybridexchange–correlationfunctionalisusuallyconstructedasalinearcombinationoftheHartree–Fockexactexchangefunctional E x HF = − 1 2 ∑ i , j ∬ ψ i ∗ ( r 1 ) ψ j ∗ ( r 2 ) 1 r 12 ψ j ( r 1 ) ψ i ( r 2 ) d r 1 d r 2 {\displaystyleE_{\text{x}}^{\text{HF}}=-{\frac{1}{2}}\sum_{i,j}\iint\psi_{i}^{*}(\mathbf{r}_{1})\psi_{j}^{*}(\mathbf{r}_{2}){\frac{1}{r_{12}}}\psi_{j}(\mathbf{r}_{1})\psi_{i}(\mathbf{r}_{2})\,d\mathbf{r}_{1}\,d\mathbf{r}_{2}} andanynumberofexchangeandcorrelationexplicitdensityfunctionals.Theparametersdeterminingtheweightofeachindividualfunctionalaretypicallyspecifiedbyfittingthefunctional'spredictionstoexperimentaloraccuratelycalculatedthermochemicaldata,althoughinthecaseofthe"adiabaticconnectionfunctionals"theweightscanbesetapriori.[2] B3LYP[edit] Forexample,thepopularB3LYP(Becke,[3]3-parameter,[4]Lee–Yang–Parr)[5]exchange-correlationfunctionalis E xc B3LYP = ( 1 − a ) E x LSDA + a E x HF + b △ E x B + ( 1 − c ) E c LSDA + c E c LYP , {\displaystyleE_{\text{xc}}^{\text{B3LYP}}=(1-a)E_{\text{x}}^{\text{LSDA}}+aE_{\text{x}}^{\text{HF}}+b\vartriangleE_{\text{x}}^{\text{B}}+(1-c)E_{\text{c}}^{\text{LSDA}}+cE_{\text{c}}^{\text{LYP}},} where a = 0.20 {\displaystylea=0.20} , b = 0.72 {\displaystyleb=0.72} ,and c = 0.81 {\displaystylec=0.81} . E x B {\displaystyleE_{\text{x}}^{\text{B}}} isageneralizedgradientapproximation:theBecke88exchangefunctional[6]andthecorrelationfunctionalofLee,YangandParr[7]forB3LYP,and E c LSDA {\displaystyleE_{\text{c}}^{\text{LSDA}}} istheVWNlocalspindensityapproximationtothecorrelationfunctional.[8] ThethreeparametersdefiningB3LYPhavebeentakenwithoutmodificationfromBecke'soriginalfittingoftheanalogousB3PW91functionaltoasetofatomizationenergies,ionizationpotentials,protonaffinities,andtotalatomicenergies.[9] PBE0[edit] ThePBE0functional[2][10] mixesthePerdew–Burke-Ernzerhof(PBE)exchangeenergyandHartree–Fockexchangeenergyinaset3:1ratio,alongwiththefullPBEcorrelationenergy: E xc PBE0 = 1 4 E x HF + 3 4 E x PBE + E c PBE , {\displaystyleE_{\text{xc}}^{\text{PBE0}}={\frac{1}{4}}E_{\text{x}}^{\text{HF}}+{\frac{3}{4}}E_{\text{x}}^{\text{PBE}}+E_{\text{c}}^{\text{PBE}},} where E x HF {\displaystyleE_{\text{x}}^{\text{HF}}} istheHartree–Fockexactexchangefunctional, E x PBE {\displaystyleE_{\text{x}}^{\text{PBE}}} isthePBEexchangefunctional,and E c PBE {\displaystyleE_{\text{c}}^{\text{PBE}}} isthePBEcorrelationfunctional.[11] HSE[edit] TheHSE(Heyd–Scuseria–Ernzerhof)[12]exchange–correlationfunctionalusesanerror-function-screenedCoulombpotentialtocalculatetheexchangeportionoftheenergyinordertoimprovecomputationalefficiency,especiallyformetallicsystems: E xc ω PBEh = a E x HF,SR ( ω ) + ( 1 − a ) E x PBE,SR ( ω ) + E x PBE,LR ( ω ) + E c PBE , {\displaystyleE_{\text{xc}}^{\omega{\text{PBEh}}}=aE_{\text{x}}^{\text{HF,SR}}(\omega)+(1-a)E_{\text{x}}^{\text{PBE,SR}}(\omega)+E_{\text{x}}^{\text{PBE,LR}}(\omega)+E_{\text{c}}^{\text{PBE}},} where a {\displaystylea} isthemixingparameter,and ω {\displaystyle\omega} isanadjustableparametercontrollingtheshort-rangenessoftheinteraction.Standardvaluesof a = 1 / 4 {\displaystylea=1/4} and ω = 0.2 {\displaystyle\omega=0.2} (usuallyreferredtoasHSE06)havebeenshowntogivegoodresultsformostsystems.TheHSEexchange–correlationfunctionaldegeneratestothePBE0hybridfunctionalfor ω = 0 {\displaystyle\omega=0} . E x HF,SR ( ω ) {\displaystyleE_{\text{x}}^{\text{HF,SR}}(\omega)} istheshort-rangeHartree–Fockexactexchangefunctional, E x PBE,SR ( ω ) {\displaystyleE_{\text{x}}^{\text{PBE,SR}}(\omega)} and E x PBE,LR ( ω ) {\displaystyleE_{\text{x}}^{\text{PBE,LR}}(\omega)} aretheshort-andlong-rangecomponentsofthePBEexchangefunctional,and E c PBE ( ω ) {\displaystyleE_{\text{c}}^{\text{PBE}}(\omega)} isthePBE[11]correlationfunctional. Meta-hybridGGA[edit] Furtherinformation:Minnesotafunctionals TheM06suiteoffunctionals[13][14]isasetoffourmeta-hybridGGAandmeta-GGADFTfunctionals.Thesefunctionalsareconstructedbyempiricallyfittingtheirparameters,whilebeingconstrainedtoauniformelectrongas. ThefamilyincludesthefunctionalsM06-L,M06,M06-2XandM06-HF,withadifferentamountofexactexchangeforeachone.M06-LisfullylocalwithoutHFexchange(thusitcannotbeconsideredhybrid),M06has27%HFexchange,M06-2X54%andM06-HF100%. Theadvantagesandusefulnessofeachfunctionalare M06-L:Fast,goodfortransitionmetals,inorganicandorganometallics. M06:Formaingroup,organometallics,kineticsandnon-covalentbonds. M06-2X:Maingroup,kinetics. M06-HF:Charge-transferTD-DFT,systemswhereself-interactionispathological. Thesuitegivesgoodresultsforsystemscontainingdispersionforces,oneofthebiggestdeficienciesofstandardDFTmethods. Despitetheirexcellentperformanceforenergiesandgeometries,wemustsuspectthatmodernhighlyparameterizedfunctionalsneedfurtherguidancefromexactconstraints,orexactdensity,orboth[15] References[edit] ^A.D.Becke(1993)."AnewmixingofHartree-Fockandlocaldensity-functionaltheories".J.Chem.Phys.98(2):1372–1377.Bibcode:1993JChPh..98.1372B.doi:10.1063/1.464304. ^abcJohnP.Perdew;MatthiasErnzerhof;KieronBurke(1996)."Rationaleformixingexactexchangewithdensityfunctionalapproximations"(PDF).J.Chem.Phys.105(22):9982–9985.Bibcode:1996JChPh.105.9982P.doi:10.1063/1.472933.Retrieved2007-05-07. ^K.Kim;K.D.Jordan(1994)."ComparisonofDensityFunctionalandMP2CalculationsontheWaterMonomerandDimer".J.Phys.Chem.98(40):10089–10094.doi:10.1021/j100091a024. ^P.J.Stephens;F.J.Devlin;C.F.Chabalowski;M.J.Frisch(1994)."AbInitioCalculationofVibrationalAbsorptionandCircularDichroismSpectraUsingDensityFunctionalForceFields".J.Phys.Chem.98(45):11623–11627.doi:10.1021/j100096a001. ^C.J.Cramer(2004)."EssentialsofComputationalChemistry:TheoriesandModels,2ndEdition|Wiley".Wiley.com.Retrieved2021-06-24. ^A.D.Becke(1988)."Density-functionalexchange-energyapproximationwithcorrectasymptoticbehavior".Phys.Rev.A.38(6):3098–3100.Bibcode:1988PhRvA..38.3098B.doi:10.1103/PhysRevA.38.3098.PMID 9900728. ^ChengtehLee;WeitaoYang;RobertG.Parr(1988)."DevelopmentoftheColle-Salvetticorrelation-energyformulaintoafunctionaloftheelectrondensity".Phys.Rev.B.37(2):785–789.Bibcode:1988PhRvB..37..785L.doi:10.1103/PhysRevB.37.785.PMID 9944570. ^S.H.Vosko;L.Wilk;M.Nusair(1980)."Accuratespin-dependentelectronliquidcorrelationenergiesforlocalspindensitycalculations:acriticalanalysis".Can.J.Phys.58(8):1200–1211.Bibcode:1980CaJPh..58.1200V.doi:10.1139/p80-159. ^Becke,AxelD.(1993)."Density-functionalthermochemistry.III.Theroleofexactexchange".J.Chem.Phys.98(7):5648–5652.Bibcode:1993JChPh..98.5648B.doi:10.1063/1.464913. ^Adamo,Carlo;VincenzoBarone(1999-04-01)."Towardreliabledensityfunctionalmethodswithoutadjustableparameters:ThePBE0model".TheJournalofChemicalPhysics.110(13):6158–6170.Bibcode:1999JChPh.110.6158A.doi:10.1063/1.478522.ISSN 0021-9606. ^abPerdew,JohnP.;KieronBurke;MatthiasErnzerhof(1996-10-28)."GeneralizedGradientApproximationMadeSimple".PhysicalReviewLetters.77(18):3865–3868.Bibcode:1996PhRvL..77.3865P.doi:10.1103/PhysRevLett.77.3865.PMID 10062328. ^JochenHeyd;GustavoE.Scuseria;MatthiasErnzerhof(2003)."HybridfunctionalsbasedonascreenedCoulombpotential".J.Chem.Phys.118(18):8207.Bibcode:2003JChPh.118.8207H.doi:10.1063/1.1564060. ^Zhao,Yan;DonaldG.Truhlar(2008)."TheM06suiteofdensityfunctionalsformaingroupthermochemistry,thermochemicalkinetics,noncovalentinteractions,excitedstates,andtransitionelements:twonewfunctionalsandsystematictestingoffourM06-classfunctionalsand12otherfunctionals".Theor.Chem.Account.120(1–3):215–241.doi:10.1007/s00214-007-0310-x. ^Zhao,Yan;DonaldG.Truhlar(2006)."DensityFunctionalforSpectroscopy:NoLong-RangeSelf-InteractionError,GoodPerformanceforRydbergandCharge-TransferStates,andBetterPerformanceonAveragethanB3LYPforGroundStates".J.Phys.Chem.110(49):13126–13130.Bibcode:2006JPCA..11013126Z.doi:10.1021/jp066479k.PMID 17149824. ^Medvedev,MichaelG.;IvanS.Bushmarinov(2017)."Densityfunctionaltheoryisstrayingfromthepathtowardtheexactfunctional".Science:215–241.doi:10.1126/science.aah5975. 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