Igor Ying Zhang and Xin Xu - Zhang, Xu - 2021 - Unknown

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pubs.acs.org/JPCLLetterExploringtheLimitsoftheXYG3-TypeDoublyHybridApproximationsfortheMain-GroupChemistry:ThexDH@B3LYPModelIgorYingZhang*andXinXu*CiteThis:J.Phys.Chem.Lett.2021,12,2638−2644ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Despitebeingthemostaccurateclassofdensityfunctionalapproximationsforthemain-groupchemistry,doublyhybridapproximations(DHAs)aregenerallyconsideredtobeincompleteindescribingthemedium-tolong-rangedispersiveinteractions.TheexistingDHAsareoftensupplementedwithempiricallong-rangedispersioncorrections.ByusingtheextensiveandchemicallydiverseGMTKN55database,weexplorethelimitsoftheXYG3-typeDHAsusingtheB3LYPreferenceorbitals,namely,xDH@B3LYP,withagraduallyrelaxedconstraintonthemixingparametersofDHAs.OurresultsdemonstratethatthexDH@B3LYPmodelcanprovideabalanceddescriptionofbothcovalentandnoncovalentinteractionswiththeaccuracyandrobustnesscomparabletoorevenbetterthantheveryexpensivecompositemethodsinwavefunctiontheory.Suchanaccuracycanbeachievedwithoutresortingtotheuseofanylong-rangecorrectionscheme,sheddingnewlightonthedevelopmentofDHAs.ThepasttwodecadeshaveseenincreasinglyrapidadvancesplayersbasedonachemicallydiverseMain-GroupChemistryinthedevelopmentofthegeneralizedKohn−ShamDatabasewithabout5000testcasesorganizedin84subsetsdensityfunctionaltheory(KS-DFT)methods.Amongthem,a(MGCDB84).Notethatthetopplayerinthisbenchmarkistherepresentativeprogressonthetop(fifth)rungofJacob’sLadderxDHofωB97M(2)whichcontains14empiricalparameters1determinedbyoptimizingtheperformancefortheMGCDB84ofKS-DFTistheinventionoftheso-calleddoublyhybrid2−19set.approximations(DHAs)inquantumchemistry.Despiteall26sharingthefeatureofusingthesecond-orderperturbativetermTheGMTKN55setoftheGrimmegroupisanotherlarge(PT2)tointroducetheunoccupiedorbitals,themainstreamdatabaseforthemain-groupchemistry.Itisacompositeof55DownloadedviaUNIVOFNEWMEXICOonMay15,2021at21:02:59(UTC).DHAscanbefurtherclassifiedaccordingtothemannerofsubsets(1505relativeenergiesintotal),representingthepreparingthedensityandorbitalsforthefinaltotal-energyGeneralMain-groupThermochemistry,Kinetics,andNon-evaluation.XYG3proposedbytheXugroupin20096definesacovalentinteractions.ItisrecommendedtouseGMTKN55forSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.familyofDHAs(xDH),20whichinsistsonmakinguseofthethefunctionaldevelopmentandbenchmarktogetherwiththeself-consistentfield(SCF)orbitalsfromaconventionallower-so-calledsecondversionoftheweightedtotalmeanabsoluterungdensityfunctionalapproximation(DFA)foragooddeviation(WTMAD2)density,whileconstructinganotherhigher-rungfunctionalfor655−1thefinalenergy.Forinstance,XYG3,itslong-range-corrected156.84kcalmol(lrc)variantlrc-XYG3,14anditsopposite-spinPT2(osPT2)WTMAD2=55∑NiMADi10∑iNii|Δ|Ei(1)variantXYGJ-OShaveallbeenestablishedbasedonthe21−24B3LYPdensityandorbitals.ThelatestxDHsinclude17where|Δ|EiistheaveragedabsoluterelativeenergyofsubsetiωB97M(2)proposedbytheHead-Gordongroupand18withthenumberofrelativeenergiesbeingNi,while56.84kcal/xrevDSD-PBEP86-D4bytheMartingroup.XYG3andXYGJ-OScontain3and4empiricalparameters,respectively.Despiteonlytheheatsofformation(HOFs)of223Received:February1,2021smallmolecules25beingusedtodeterminetheparameters,Accepted:March4,2021XYG3andXYGJ-OShavebeendemonstratedtobecompetentPublished:March10,2021forvariouskindsofchemicalinteractionsofthemain-groupchemistry.InaveryrecentcomprehensivebenchmarkbytheHead-Gordengroup,XYG3andXYGJ-OSareboththetop-10©2021AmericanChemicalSocietyhttps://dx.doi.org/10.1021/acs.jpclett.1c003602638J.Phys.Chem.Lett.2021,12,2638−2644

1TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterTable1.EmpiricalParametersforthexDH@B3LYPFunctionals,TheirWTMAD2ValuesontheGMTKN55Benchmark,and28TheirMADValuesfor142CalculatedBondDissociationEnergiesofSmallMolecules(BDE142)(inkcal/mol)aparametersxDH@B3LYPa1a2a3a4a5a6a7a8a9WTMAD2BDE142XYG30.8033[−0.0140]0.2107[0.0000][0.6789]0.3211[0.3211]−−3.742.10lrc-XYG30.8033[−0.0140]0.2107[0.0000][0.6789]0.3211[0.3211][0.4822][0.4822]3.031.84XYGJ-OS0.7731[0.2269][0.0000]0.23090.27540.4364[0.0000]−−4.010.86lrc-XYGJ-OS0.7731[0.2269][0.0000]0.23090.27540.4364[0.0000]1.0300[0.0000]3.010.89revXYGJ-OS0.8877[0.1123][0.0000]−0.06970.61670.5485[0.0000]−−3.022.20XYG-OS50.8928[0.3393]−0.23210.3268−0.06350.5574[0.0000]−−2.661.56revXYG30.9196[−0.0222]0.1026[0.0000][0.6059]0.3941[0.3941]−−2.471.34XYG50.9150[0.0612]0.0238[0.0000]0.49570.45480.2764−−2.321.13XYG60.9105[0.1576]−0.06810.18000.22440.46950.2426−−2.241.09XYG70.89710.2055−0.14080.40560.11590.40520.2589−−2.051.03aDetaileddescriptionoftheparametersandtheassociatedDFTcomponentsareprovidedineqs2and3andtherelevantparagraphs.Tobespecific,inthe(lrc-)xDH@B3LYPmethods,{a,a,a}aretheparametersfortheexchangecomponentsof{EHF,ES,EB88},respectively,and{a,123xxx4a}forthe(semi)-localcorrelationcomponentsof{EVWN,ELYP}.{a,a}aretheparametersfortheOSandSSpartsofthePT2correlation{EosPT2,5cc67cEssPT2},while{a,a}areforthecorrespondinglong-rangecontributions{Eos‑lrPT2,Ess‑lrPT2}.Parameterswithsquareblacketsaretheconstrainedc89ccparameters,determinedbytheconstraintsofCnwithn=1−9.molistheaverageofall55{|Δ|}Ei.MADireferstothemeanorbital(NAO)basissetwithvalencecorrelationconsistencyinabsolutedeviationofsubseti.GMTKN55withWTMAD2thequadruple-ζquality(NAO-VCC-4Z)fortheelementsfrom1HtoAr30andtheGaussian-typebasissetofcc-pwCVQZfornumericallyvalidatestheconceptofJacob’sLadderofKS-DFT31thatahigher-rungDFAcanbemoreaccuratethanalower-rungheavierelements,whicharetabulatedwithatighternumericalDFA.Thereisnoreporttodateontheperformanceof(lrc-integrationsettinginFHI-aims.ForthesubsetsofWATER27,)XYG3andXYGJ-OSfortheGMTKN55database.However,aRG18,IL16,G21EA,andAHB21,theaforementionedbasissets31,32recentbenchmarksuggeststhatthexDHmethodofωB97M(2),areaugmentedwithdiffusefunctionsfromaug-cc-pVQZ.thetopplayerfortheMGCDB84database,isalsothewinnerforThefrozen-coreapproximationisadoptedforthePT2GMTKN55(WTMAD2=2.19kcal/mol).18ThexrevDSD-evaluationswithonlyvalenceelectronscorrelatedformostofPBEP86-D4methodhas6empiricalparameters,whichweretheelements.TheelectronsonthesecondvalenceshellareoptimizeddirectlyagainstthefullGMTKN55database,yieldinginvolvedonlyforthealkali(-earth)elements.Notethatthe18,26aWTMAD2of2.23kcal/mol.18ThesetwoxDHmethodspreviousGMTKN55benchmarksusethedef2-QZVPPbaisrepresentthetop-levelaccuracyofKS-DFTcurrentlyavailablesetwithasimilarchoiceofaugmenteddiffusebasisfunctionsandforthemain-groupchemistry,27whicharecomparabletothefrozen-coreapproximationsforspecificelementsandsubsets.muchmoreexpensivecompositemethodsinwavefunctionTheoveralldifferenceassociatedwiththeabovenumericaltheory(WFT),likeCBS-QB3(2.25kcal/mol),G4MP2(2.29settinghasledtoadeviationofWTMAD2around0.05kcal/molfortheB3LYPmethod,althoughthedifferenceforaspecifickcal/mol),andG3B3(2.20kcal/mol).TheWTMAD2valuesofsubsetcouldbeslightlyhigherasshowninTable2.theWFTmethodswereobtainedwithareducedsetof642Tobetterstudytheefficacyoftheparametrizationinthereactionenergies,forwhichthecomputationalcostswere27xDH@B3LYPmodel,wecollectthem,includingXYG3andaffordable.XYGJ-OS,inthefollowinggeneralformfortheexchange-Inthisletter,webenchmarktheaccuracyofXYG3andXYGJ-correlationenergyOSforthemain-groupchemistryusingtheGMTKN55database.Afterward,reparametrizationundertheoriginalxDH@B3LYPHFSB88VWNformulas(yieldingrevXYG3andrevXYGJ-OS)iscarriedoutEaxc=+++1Exa2Exa3Exa4EcbydirectlyminimizingtheWTMAD2ofthefullGMTKN55+++aELYPaEosPT2aEssPT25ccc67(2)database.WethenexplorethelimitsofthexDHmethodsusingB3LYPdensityandorbitals(xDH@B3LYP)bygraduallywhereEHFisthecontributionfromtheHartree−Fock(HF)-likexrelaxingtheconstraintsintheoriginalformula.ThenewxDHexactexchange,whileESandEB88aretheexchangecontributionsxxmethodsincludeXYG5,XYG-OS5,XYG6,andXYG7withthefromtheSlater-typelocaldensityapproximation(LDA)33andlastcharacterbeingthenumberoftheempiricalparameters.The21theBecke88generalizedgradientapproximation(GGA),optimizedparametersofthesemethodsarelistedinTable1.VWN34respectively.EcisthelocalVosko−Wilk−Nusaircorrela-OurresultsconfirmthattheoriginalXYG3andXYGJ-OSaretion.Forconsistency,thexDH@B3LYPmodelutilizesthesamecapableofdeliveringasatisfactorydescriptionofthe24VWNversionadoptedintheoriginalB3LYPimplementation.GMTKN55benchmark,whilereparametrizationwithanELYPistheLee−Yang−Parr22correlationapproximation.EosPT2ccincreasingnumberofparametersactuallyimprovestheirgeneralisthefifth-rungcorrelationcontributionintheformofopposite-performances.Surprisingly,suchanaccuracycanbeachievedspinPT2(osPT2),andEssPT2denotesthesame-spinpartofPT2cwithoutresortingtotheuseofempiricaldispersionsorlong-(ssPT2).rangePT2corrections.Thegeneralformula(eq2)ofthexDH@B3LYPmodelhas7CalculationsareallperformedusingtheFritz−Haberempiricalparametersintotal.XYG3imposes4constraintsastheInstituteabinitiomolecularsimulations(FHI-aims)packagefollowing,suchthatonly3parametersremain,29inthenumericalintegrationframework.FormostsystemsintheGMTKN55database,weusethenumericalatomic-centera123++=aa1.0(C1)2639https://dx.doi.org/10.1021/acs.jpclett.1c00360J.Phys.Chem.Lett.2021,12,2638−2644

2TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterTable2.WTMAD2ValuesofDifferentDFAs(inkcal/mol)associatedwiththebasicpropertiesofthereactionsofsmallfortheFullGMTKN55Benchmark(All),BasicPropertiessystems(Sub1with18subsets),isomerizationandreactionsofandReactionsofSmallSystems(Sub1),Isomerizationsandlargesystems(Sub2with9subsets),barrierheights(Sub3,7ReactionsofLargeSystems(Sub2),BarrierHeights(Sub3),subsets),andintermolecular(Sub4,12subsets)andintra-IntermolecularNoncovalentInteractions(Sub4),andmolecular(Sub5,9subsets)noncovalentinteractions(pleasearefertotheSIformoredetailedMADsofthe55subsets).InlineIntramolecularNoncovalentInteractions(Sub5)17,30,37,38withpreviousstudies,theGMTKN55benchmarkallSub1Sub2Sub3Sub4Sub5confirmsthatXYG3andXYGJ-OSarequiteaccuratefortheB3LYP16.235.2617.539.1928.2325.14main-groupchemistrybutstillhaveroomforfurtherimprove-bB3LYP16.185.1017.269.0428.5725.11mentforsystemsinvolvingthenoncovalentinteractions(e.g.,XYG33.751.724.692.236.044.88Sub2,Sub4,andSub5).AsshowninTable2,XYG3andXYGJ-lrc-XYG33.031.644.152.434.912.78OSarecomparabletothebestexistingDFA,ωB97M(2)forXYGJ-OS4.041.854.422.946.015.97smallsystems(Sub1),andbarrierheights(Sub3).Buttheirlrc-XYGJ-OS3.011.793.812.624.732.80WTMAD2valuesforSub4andSub5aretwotothreetimesrevXYGJ-OS3.022.343.563.094.122.46largerthanthoseofωB97M(2).XYG-OS52.661.893.413.263.202.35ItisnowwidelyacceptedthatthedoublyhybridschemeitselfrevXYG32.471.592.862.743.302.56cannotfullycoverthemedium-tolong-rangecorrelationforXYG52.321.552.592.513.152.33noncovalentinteractions.Hencethelong-rangecorrectionXYG62.241.472.512.612.892.31strategies,suchastheD3dispersionusedinB2PLYP-D3andXYG72.051.312.432.302.782.00DSD-BLYP-D3(thesuffix“D3”istheabbrevationof“D3(BJ)”bB2PLYP-D33.932.516.284.903.783.78forsimplicitythroughoutthepaper),theD4dispersioninbDSD-BLYP-D33.081.884.323.043.923.15xrevDSD-PBEB86-D4,andthenonlocalVV10dispersioncωB97M(2)2.191.412.581.992.452.99employedinωB97M(2),areoftennecessarytoassistthedxrevDSD2.231.803.032.112.332.23DHAstowardabalanceddescriptionofcovalentandnon-aThebesttwoDFAswiththesmallestWTMAD2valuesforthecovalentinteractions.In2013,weproposedadispersionbwholesetandeachsubsetaremarkedinbold.Resultsfromref26correctionschemebasedonthelong-rangepartofPT2c14usingthebasissetofdef2-QZVPP.Resultsfromref18usingthe(lrPT2)tobetteraccountfortheorbitaldependence,indbasissetofdef2-QZVPP.AcronymofxrevDSD-PBEB86-D4;resultsparticular,ofthemedium-rangecorrelation.Forthesakeoffromref18aswell.consistency,werecastthelong-rangecorrectedXYG3(lrc-XYG3)inagenerallrc-xDH@B3LYPformulaasa4=0.0(C2)lrcxDH@B3LYP‐‐xDH@B3LYPoslrPT2sslrPT2‐a56+=a1.0(C3)EEaxc=++xc8Eca9Ec(3)a67=a(C4)os‑lrPT2ss‑lrPT2osPT2ssPT2Ec(orEc)sharesthesameformulaasEc(orEc)Thefirstandthirdconstraints(C1andC3)arethebutdecoratestheelectron−electronrepulsiveoperatorbyanormalizationconditionsfortheexchangeandthecorrelationGausserrorfunction,v̂lr=erf(ωr)/r,wheretheparameterωee1212contributions,respectively.Thesecondconstraint(C2)determinesthedistanceatwhichtheasymptoticlimitv̂lr→v̂=eeeeexcludestheuseofEVWNfortheparametrization,andthefourth1/risreached.Intheoriginalpaper,14ω=0.2Bohr−1wasc12constraint(C4)indicatesthatthePT2correlationEPT2=EosPT2ccadoptedasbeingappropriateaccordingtoref3,whichisalso+EssPT2isconsideredasaunitfortheparametrization.cusedinthisstudyforalllrc-xDH@B3LYPmethods.Similarly,eq2reducestotheXYGJ-OSformulawith4lrc-XYG3doesnotintroduceamoreempiricalparameterbutparametersbyintroducing3constraintsasinsteadhasanextraconstraint(C8),a12+=a1.0(C5)a8916==−aaa(C8)a3=0.0(C6)whichregulatesthePT2correlationtohavethesameamountoftheHFexchangeinthelong-range,whileretainingalltheothera7=0.0(C7)parametersasintheoriginalXYG3fittedonlyagainsttheHOFsofsomesmallmoleculesintheG3set.ThepresentstudyXYGJ-OSalsonormalizestheexchangeenergiesviathefifthB88confirmsthatthelong-rangecorrectionnotablyimprovestheconstraint(C5),butexcludestheuseofExfortheperformanceofXYG3,reducingtheWTMAD2from3.75toparametrization(thesixthconstraint,C6).Theseventh3.02kcal/mol,whichisnowcomparabletotheperformanceofconstraint(C7)wasoriginallyintroducedtoavoidthetheDSD-BLYP-D3method(WTMAD2=3.08kcal/mol),theevaluationofthessPT2contribution,suchthatthecomputa-5bestDFAoutof217candidatesintheoriginalGMTKN55tionalscalingcanbereducedfromtheorderofN(Nmeasuring264benchmarkwork.Wethenproposeinthisletteralong-rangethesystemsize)toNifusingtheresolution-of-identity35,36correctionforXYGJ-OSusingasimilarstrategyastoconstructexpansionswiththeLaplacequadratureapproximations,lrc-XYG3butwithonlytheos-lrPT2contribution:whichcanbefurtherreducedtoaquasi-linearscalingwiththe10VWNaidofalocalimplementationofosPT2.Inconsequence,Eca9=0.0(C9)withparametera4isreintroducedtocompensatethemissingssPT2contribution.andanextraparameterofa8=1.0300.Akintothelrc-XYG3Inadditiontotheempiricalparametersandtheconstraintsofperformance,lrc-XYGJ-OSyieldsanimprovedWTMAD2ofXYG3andXYGJ-OS,Table1liststheirWTMAD2valuesforthe3.01kcal/molwithasimilaraccuracyoflrc-XYG3andDSD-GMTKN55benchmark,whileTable2providesthebreakdownBLYP-D3.CloserinspectionofTable2showsthatsuchan2640https://dx.doi.org/10.1021/acs.jpclett.1c00360J.Phys.Chem.Lett.2021,12,2638−2644

3TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterFigure1.Normalizedmeanabsolutedeviation(NMAD,unitof%)asthepercentageofaDFA’sMADagainsttheaveragedabsoluterelativeenergy|ΔE|for55subsetsintheGMTKN55database.ResultsofrevXYG3,XYG6,andXYG7arecomputedinthepresentwork,whilethoseofDSD-BLYP-18,26D3,ωB97M(2),andxrevDSD-PBEB86-D4areobtainedfromthepreviousliterature.improvementischieflyassociatedwithabetterdescriptionofemployinganyspecifictreatmentofthelong-rangecorrelationsnoncovalentinteractions.fornonbondedinteractions.Suchapositiveroleplayedbythelong-rangecorrectionLetusnowturntothesubstitutabilityofthessPT2correlationdemonstratesthatthechiefflawofXYG3andXYGJ-OSfortheinxDH@B3LYP.ItisanattractivetopicforanyPT2-basedgeneralmain-groupchemistryindeedoriginatesfromtheDHAs,sincealower-scalingimplementationisavailableifonly10,35,36deficiencyindescribingthenoncovalentinteractions.However,theosPT2correlationisconsidered.TheXYG-OS5suchanobservationitselfmaynotbeadirectsupportofthemethodisestablishedbyimposingonly2constraintsofC1andaforementionedconsensus:thedoublyhybridschemeitselfcannotC7tothegeneralxDH@B3LYPformula(eq2).BothXYG5andfullycoverthemedium-tolong-rangecorrelationfornoncovalentXYG-OS5contain5parameters.Table1showsthattheonlyinteractions.differenceisthatthesame-spincorrelationisexclusivelyconsideredbyEssPT2inXYG5,whichissubstitutedbytheGiventhefactthattheparametrizationoftheoriginalXYG3cLDAcorrelationofEVWNinXYG-OS5.Table2suggeststhatandXYGJ-OSemployedonlytheHOFsofsomesmallcmoleculesintheG3setasthetrainingset,whichmayfavorsuchakindofsubstitutionsomewhatdowngradesthecovalentinteractions,whiledisfavoringnoncovalentinterac-performancequiteconsistently,leadingtoanoveralltions,weperformareparametrizationofXYG3andXYGJ-OSWTMAD2of2.32kcal/molfortheformerto2.66kcal/molagainstthefullGMTKN55database.AsshowninTable1,theforthelatterforthewholeGMTKN55benchmark.Nonethe-revisedXYGJ-OS(i.e.,revXYGJ-OS)improvestheaccuracyforless,suchasubstitutionisareasonablecompromiseintermsofthemain-groupchemistryovertheoriginalXYGJ-OSbyabout1efficiency.kcal/mol,descreasingWTMAD2to3.02kcal/mol.MoreThesmallestWTMAD2(2.05kcal/mol)givenbyXYG7issurprisingly,therevisedXYG3(i.e.,revXYG3)withonly3outstanding,althoughitdoesnotnecessarilyindicatethatXYG7parametersisfoundtodeliveranexcellentperformancefortheistheoverallwinnerforthemain-groupchemistry.Thedisparitymain-groupchemistry.TheWTMAD2ofrevXYG3isonly2.47oftheWTMAD2valuesislessthan0.2kcal/molamongXYG6,kcal/mol,approachingtothoseofthetop-classstate-of-the-artXYG7,ωB97M(2),andxrevDSD-PBEB86-D4,whichputtheseDHAs,suchasxrevDSD-PBEB86-D4with6parametersandDHAsasthetop-classperformers.Meanwhile,itisalwaysωB97M(2)with14parameters.worthwhiletounpacktheWTMAD2valuesandlookintotheirWethenexploretheperformanceofthexDH@B3LYPmodelperformanceforanindividualsubset,whichisimportanttoagainstthenumberofparameters.XYG5introduces2morejudgetheirrobustnessandtogivemorepreciseadvicetotheparametersbyremovingthenormalizationofthecorrelationuserswhomightbeinterestedinaspecificproblemcloselyenergies(C3)andgeneralizingthePT2modeltothespin-relatedtooneorseveralsubsetsinthebenchmark.Inthisregard,componentscaledPT2(scs-PT2)model(C4).39,40XYG6weprovidetheMADsofallDHAsinvestigatedforeachfurtherintroducestheLDAcorrelationofEVWN(C2)andXYG7individualsubsetintheSI,whileFigure1showsthenormalizedcultimatelyunlocksall7parametersinthemostgeneralform(eqMADi(NMADi)asthepercentageofMADiagainstthe2)ofthexDH@B3LYPmodel.Table1liststhecorrespondingaveragedabsoluterelativeenergy|Δ|EiforsixselectedDHAs.AsoptimizedparametersandtheirfinalWTMAD2valuesfortheillustratedinFigure1,allsixDHAsdeliverquiteasimilarwholeGMTKN55dataset.FromthedatainTable1,apositiveperformanceformostofthesubsetswiththerespectiveMADicorrelationisfoundbetweenthenumberofparametersandthebeingsmallerthan5%of|Δ|Ei.However,divergenceexistsinfinalperformance.StatisticsforthefivesubcategoriesinTable2severalspecificproblems.Forexample,theSIE4×4subsetdemonstratetheconspicuousimprovement,whichishighlycontains16dissociationenergiesoffourpositivelychargedconsistentnotonlyforthefullGMTKN55databasebutalsofordimersatfourdifferentpointsalongtheirdissociationpotentialdifferentkindsofchemicalinteractions.Thisfindingisclearlyenergycurves.Itisatypicaltestsetfortheself-interaction-errorcontrarytothegeneralopinionaforementioned,andweactually(SIE)problems,whereourxDH@B3LYPmodelexhibitsafindthatbothcovalentandnoncovalentinteractionsofsmallmuchbetterperformance.ItcanbeattributedtothelargeandlargesystemscanbeproperlyandverysatisfactorilyamountofEHFusedintheenergyfunctionalofxDH@B3LYP,xdescribedinthecontextofxDH@B3LYPwithoutresortingtowhichisanaturaloutcomeoftheparametrizationagainstthe2641https://dx.doi.org/10.1021/acs.jpclett.1c00360J.Phys.Chem.Lett.2021,12,2638−2644

4TheJournalofPhysicalChemistryLetterspubs.acs.org/JPCLLetterentireGMTKN55database.ThexDHschemeallowsforrandomnoisewithin±0.25(“random.uniform”)foreachdifferentamountsofEHFusedintheSCFfunctionalandtheparameter.Ourparametrizationprocedureensuresafullxenergyfunctional,offeringaneffectivewaytominimizetheoptimizationtofindtheglobalminimumandsolidifiesournotoriousSIE.Allinall,XYG7ishighlyrobustwithnosingleresultsandconclusionagainsttheoptimizationincompleteness.NMADibeinglargerthan15%.ThelargestNMADiofXYG7isThelatestxDHoftheHead-Gordongroup,ωB97M(2),1711.7%fortheTAUT15subset,whichhasasmall|Δ|EiofonlyemploysthedensityandorbitalsfromωB97M-V,whichisthe183.05kcal/mol,resultinginafairlysmallMADof0.36kcal/mol.bestexistinghybridDFAfortheGMTKN55benchmark.TheBesidestheGMTKN55database,wecalculate142bondPT2coefficientofωB97M(2)is0.34096,closetotheoptimizeddissociationenergies(BDE142)ofsmallmolecules,28asBDEisainXYG3andrevXYG3.Duetotheconstraintusedto6thecentralconceptinchemistry,whiletheBDE142setisnotnormalizethecoefficientsofthePT2correlationandtheVV10includedintheparametrizationofthexDH@B3LYPmethods.dispersion(seeeq8inref17),itbringsinalargeportion(aboutTable1showsthatthebestperformanceisgivenbyXYGJ-OS,65%)oftheVV10dispersiontotheωB97M(2)totalenergy.AsyieldinganMADofonly0.86kcal/mol.Reparametrizationofdiscussedintheoriginalpaper,theauthorsofωB97M(2)alsoXYGJ-OS(revXYGJ-OS)deterioratestheperformance,increas-triedtheoptimizationwithouttheaforementionedconstrainoningtheMADforBDE142to2.20kcal/mol.ThismaysuggestthePT2correlationandtheVV10dispersion,which,however,17thattheformulaforthefunctionalistooconstrained.Indeedtheproducesaworseperformance.Inprinciple,afulloptimizationaccuracyoftheosPT2-basedxDH@B3LYPmethodisimprovedwithmoreparametersshouldleadtoabetterperformance.Inwiththeincreasingnumberofparameters.HencetheMADoftheworstcase,asencounteredinouroptimizationofthelrc-XYG-OS5fortheBDE142setisaround1.56kcal/mol.WealsoxDH@B3LYPmethods,itatleastshouldproducethesameobserveapositivecorrelationbetweentheperformanceandtheperformanceofthepreviousoptimization,indicatingthenumberofparametersfromtheMADsofrevXYG3,XYG5,redundancyoftheextraparameters,likea8,a9inourXYG6,andXYG7downto1.0kcal/molfortheBDE142testset,optimizationforthelong-rangePT2contributions(seeeq3).arguablyreachingtheremaininguncertaintiesinthereferenceAccordingtothefindinginthiswork,weexpectthatasimilardata.Moreover,FiguresS1andS2intheSIsuggestthatourperformancecouldbeachievedbyabolishingtheconstraintandxDH@B3LYPmodelcanprovideasatisfactorydescriptioninevenremovingtheVV10dispersionfortheoptimizationofdescribingthewholeH+CH4→H2+CH3reactionpathandωB97M(2).theinteramolecularpotentialofthemethane−benzenecomplex,Insummary,acomprehensiveandsystematicstudybasedonbothofwhicharenotincludedintheparametrizationaswell.AlltheGMTKN55database,anextensiveandchemicallydiversetheseobservationsfurtherdemonstratethatthexDH@B3LYPdatabaseforthemain-groupchemistry,hasbeencarriedouttomodelisconvincinglyrobustforthegeneralmain-groupproposeandbenchmarkaserialoftheXYG3-typeDHAsthatchemistry.useB3LYPdensityandorbitals,namely,xDH@B3LYP.OurTofurtherstudythepossiblebenefitofthelong-rangeresultssuggestthatthexDH@B3LYPmodeliscapableofcorrections,weemploythegenerallrc-xDH@B3LYPformulaprovidingabalancedandhighlyaccuratedescriptionofboth(eq3)toreparametrizethelrcextensionofrevXYG3andXYGncovalentandnoncovalentinteractionsofsmallandlargewithn=5,6,7againstthefullGMTKN55database.Thesystems,whichoffertheoverallbestaccuracyforthegeneralparametrizationof(a8and/ora9)isaccompaniedbytheothermain-groupchemistry.Moreinterestingly,suchanaccuracyisparameterseitherlockedoroptimizedsimultaneously.Itisachievedwithoutresortingtotheuseofempiricaldispersionorsomewhatsurprisingthatnobenefitofusingspeciallong-rangelong-rangePT2corrections.correctionstopromotexDH@B3LYPcouldbeidentifiedinthisItisnowwellrecognizedthataDFAcontainstwosourcesof41,42analysis.Inparticular,thefullparametrizationof(a1,a3,a6)inerrors:oneisdensity-driven,andtheotherisenergy-driven.XYG3pluslrc(a8witha9=a8)leadstoasetofparametersthatInthexDHscheme,thetaskstoeliminatethesetwoerrorscanareidenticaltothoseofrevXYG3withanegligiblea8lessthanbepursuedseparatelyandsequentialybyoptimizingtheSCF0.005.Theoptimizeda8and/ora9forXYGnareintheorderoffunctionalandthentheenergyfunctionalseparately.Originally,0.01withatinyadjustmentofotherparametersinXYGn.TheweadoptedB3LYPastheSCFfunctionalforXYG3andXYGJ-resultinglrc-XYGnwithn=5,6,7yieldsalmostidenticalOS,astheB3LYPdensitieswerefoundtobesimilartothoseWTMAD2valuesasthestandardXYGnnotonlyforthefull43fromhigh-levelWFTforthemoleculesstudied.ItcouldbeGMTKN55setbutalsoforthefivesubcategories.Con-bettertousesomedifferentsetoforbitals,forexample,thePBE0sequently,wedonotreporttheselrcvariantsandtheirresults13referenceusedforxDH-PBE0,theωB97M-VreferenceforinTables1and2.Theredundancyofthelong-rangecorrections17ωB97M(2),andothers.NonlocalcorrelationotherthanthetothexDH@B3LYPmodelfurtherconfirmsthecapabilityofthePT2term,forexample,therandomphaseapproximationxDH@B3LYPframeworkitselftodeliverabalanceddescription(RPA),44−49canalsobeexploredasameanstoextendtheofbothcovalentandnoncovalentinteractionsofsmallandlargeapplicabilityofDHAsbeyondthemain-groupchemistry.systems.ThefactthatthexDH@B3LYPmodelusesthesameB3LYP■densityandorbitalsdramaticallyreducestheparametrizationASSOCIATEDCONTENTdifficulty,becausethetime-consumingSCFandpost-SCF*sıSupportingInformation(lr)PT2evaluationsarerequiredonlyonceinthepreparationofTheSupportingInformationisavailablefreeofchargeateachexchangeandcorrelationcomponentsineqs2and3,https://pubs.acs.org/doi/10.1021/acs.jpclett.1c00360.deservestobementioned.OurhomemadepythonscriptingDetaileddescriptionofthetestset,computationaldetails,employsthedownhillsimplexalgorithm(“scipy.optimize.fmin”)calculationresultsofvariousDFAsforbothGMTKN55tosearchthelocalminimumforagiveninitialguess.ThelocalandBDE142datasets,variouspotentialenergycurvesforminimizationisrepeatedmorethan100timeswiththeinitialguessinheritedfromtheprevioussearchbutperturbedbyatheH+CH4→H2+CH3reaction,andthe2642https://dx.doi.org/10.1021/acs.jpclett.1c00360J.Phys.Chem.Lett.2021,12,2638−2644

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