Electroncorrelationinmolecules--abinitio beyondGaussianquantumchemistry1stEdition Hoggan
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Stop People Pleasing: How to Start Saying No, Set Healthy Boundaries, and Express Yourself Chase Hill
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FrankJensen(Aarhus,Denmark)
MelLevy(Greensboro,NC,USA)
JanLinderberg(Aarhus,Denmark)
WilliamH.Miller(Berkeley,CA,USA)
JohnW.Mintmire(Stillwater,OK,USA)
ManojMishra(Mumbai,India)
JensOddershede(Odense,Denmark)
JosefPaldus(Waterloo,Canada)
PekkaPyykko(Helsinki,Finland)
MarkRatner(Evanston,IL,USA)
DennisR.Salahub(Calgary,Canada)
HenryF.SchaeferIII(Athens,GA,USA)
JohnStanton(Austin,TX,USA)
HarelWeinstein(NewYork,NY,USA)
isillustratedbyGrabowski etal. inrecentworkonthenoblegasdimers, evaluatingcorrelationeffectsindensityfunctionaltheoryalongthedissociationpath.Early2014alsosawthefirstvalidated abinitio benchmarkforthe H2 dissociationenergybarrieronCu(111),givenusingQMCbyHoggan: 14.79kcal/mol,wellwithinstandarderror(0.5)oftheexperimentalvalue 14.48kcal/mol.(seeArXiv).
Acoupleofnovelresearch-reviewsclosethisvolume,wheretheoryis usedasaverydirectcomplementaryapproachtoexperiment,oneon X-RayConstrainedWaveFunctions:FundamentalsandEffectsofthe MolecularOrbitalsLocalizationbyGenoniandtheotherregardingElectron ImpactIonizationbySaha etal.
Hopedfullyreadersenjoythisvolumeasmuchaswehaveenjoyed editingit!
PHILIP E.HOGGANAND TELHAT OZDOGAN Editors
SeldaAkdemir
DepartmentofScienceEducation,FacultyofEducation,SinopUniversity,Sinop,Turkey
LorenzoUgoAncarani
EquipeTMS,UMRCNRS7565,ICPM,Universite ´ deLorraine,57078Metz,France
RolandAssaraf
SorbonneUniversite ´ s,UPMCUnivParis06,andCNRS,UMR7616,Laboratoire deChimieThe ´ orique,Paris,France
ArunK.Basak
DepartmentofPhysics,UniversityofRajshahi,Rajshahi,Bangladesh
AhmedBouferguene
Faculte ´ Saint-Jean,UniversityofAlberta,Alberta,Canada
AdamBuksztel
InstituteofPhysics,FacultyofPhysics,AstronomyandInformatics,NicolausCopernicus University,Torun,Poland
MelekEraslan
DepartmentofPhysics,FacultyofArtsandSciences,AmasyaUniversity,Amasya,Turkey
MuratErturk
DepartmentofPhysics,Facultyofartsandsciences,OnsekizMartUniversity,C¸anakkale, Turkey,andInstituteofPhysics,NicholasCopernicusUniversity,Torun,Poland
AlexeiM.Frolov
DepartmentofAppliedMathematics,UniversityofWesternOntario,London,Ontario, Canada
GustavoGasaneo
DepartamentodeFı´sica,UniversidadNacionaldelSur,8000Bahı´aBlanca,BuenosAires, andConsejoNacionaldeInvestigacionesCientı´ficasyTe ´ cnicasCONICET,Argentina
DanielH.Gebremedhin
DepartmentofPhysics,FloridaA&MUniversity,Florida,USA
AlessandroGenoni
CNRS,andUniversite ´ deLorraine,LaboratoireSRSMC,UMR7565, Vandoeuvre-les-Nancy,France
NikitasI.Gidopoulos
DepartmentofPhysics,DurhamUniversity,Durham,UnitedKingdom
IreneuszGrabowski
InstituteofPhysics,FacultyofPhysics,AstronomyandInformatics,NicolausCopernicus University,Torun,Poland
CharlesA.Weatherford DepartmentofPhysics,FloridaA&MUniversity,Florida,USA
JunYasui
RIKENInnovationCenter,Hirosawa,Wako,Saitama,andFrontierResearchCenter, CanonInc.,Ohta-ku,Tokyo,Japan
NiyaziYu ¨ kc ¸ u ¨ DepartmentofEnergySystemsEngineering,FacultyofTechnology,AdıyamanUniversity, Adıyaman,Turkey
MagdalenaZientkiewicz FacultyofPhysics,UniversityofWarsaw,Warsaw,Poland
CarlosMarioGranados-Castro*,†,1,LorenzoUgoAncarani*,1 , GustavoGasaneo†,{,DarioM.Mitnik{,}
*EquipeTMS,UMRCNRS7565,ICPM,Universite ´ deLorraine,57078Metz,France
†DepartamentodeFı´sica,UniversidadNacionaldelSur,8000Bahı´aBlanca,BuenosAires,Argentina {ConsejoNacionaldeInvestigacionesCientı´ficasyTe ´ cnicasCONICET,Argentina
}InstitutodeAstronomıayFısicadelEspacio(IAFE)andDepartamentodeFısica,UniversidaddeBuenosAires, C1428EGABuenosAires,Argentina
1Correspondingauthors:e-mailaddress:carlos.mario-granados.castro@univ-lorraine.fr; ugo.ancarani@univ-lorraine.fr
structureofthemoleculecanbeignored,especiallyinhigh-energycollisions,justifyingtheuseoftheBorn–Oppenheimer(BO)approximation. Also,inordertosimplifythecalculations,thefrozencore(FC)approximationandthestaticexchangeapproximation(SEA)areconsidered.Itis withinthisframe,togetherwithamodelmolecularpotential,thatweimplementthegeneralizedSturmianapproach.Intheliterature,severalSturmian functionimplementationsexist,asreviewed,e.g.,intheintroductionsof Refs. 22 and 23.Similarlytopreviouspublicationsonscatteringstudies (see the recentreview 22 andreferencestherein),inthiscontributionwe shallname Generalized SturmianFunctions (GSF)thosedefinedin Section5.1;notethatotherauthorsusethesameterminologytodefinea different class ofSturmianfunctions.Oneoftheadvantagesofsuchamethod isthatitensuresthatthecontinuumwavefunctionhasthecorrectasymptotic behavior.22 Toassessthevalidityofourapproach,wewillcomparethecalculatedPIcrosssectionsforanumberofsmallmoleculeswiththeoretical andexperimentaldatafoundintheliterature.
Therestofthispaperisorganizedasfollows.WestartwithsomegeneralitiesonPIin Section2;wecontinuein Section3 withabriefpanorama of what sortofagreementoneobservesintheliteraturebetweentheoretical andexperimentalcrosssections.In Section4,wepresentasurveyofexisting theoretical methods usedtoinvestigatemolecularPI.In Section5,weintroducethe Sturmianapproach,andcompareourresultsforPIofH2O,NH3, andCH4 toseveraltheoreticalandexperimentaldata.
Atomicunits(ℏ ¼ e ¼ me ¼ 1)areassumedthroughout,unlessstated otherwise.
Inthestudyoftheinteractionofaradiationfield(aphoton)witha moleculartargetseveralprocessesmayoccur.Consideraphotonofenergy Eγ ¼ ℏω,suchthat Eγ > I0,where I0 istheionizationpotentialofthemolecule.OnceitstrikesapolyatomicmoleculeRAinaninitialvibrationalstate ν0 (RisthepolyatomicradicalandAisanindividualatom),thedifferent outcomesmaybe
Ifwehaveadissociationprocess,thefinalproductscanbeinanexcitedstate. Ifwehaveanejectedelectron,calledphotoelectron,ithasadefinedangular momentum ‘.Inthiscontribution,wewillconcentrateonlyonsinglePI whichcanbeconsideredasa“half-scattering”processes.Itinvolvesa bound-freetransitionforwhichoneneedstoknowonlytheinitialstate Ψ0 (energy E0)ofthemolecule,usuallyitsgroundstate,andthefinalstate oftheionizedelectron.Thetransitionoperator,thatconnectsbothinitial andfinalstates,isdescribedsemi-classicallyviathedipolarapproximation; thedipolaroperatorinbothlength(L)andvelocity(V)gaugesreads
where ^ ε givesthepolarizationofthefield.Inthiswork,weconsiderlinear polarizationalongthe z direction.
Themajortaskistocalculateaccuratelythewavefunction Ψ ofthephotoelectron,thatisanelectroninacontinuumstateoftheionizedmolecular target,withanenergy E ¼ k2/2definedbytheenergyoftheincidentphoton E ¼ Eγ I0.Suchcontinuumwavefunctionsaremoredifficulttocalculate thanthelow-lyingbound-statesastheyoscillateuptoinfinity.Theyare solutionsofthetime-dependentSchr € odingerequation(TDSE)orthe time-independentSchrodingerequation(TISE),withwelldefinedproperties.Theymustberegularattheoriginofthecoordinatesystem,andthe asymptoticboundaryconditionsaregivenbythesuperpositionofan incoming-waveCoulombfunctionplusanincomingsphericalwave, generatedbythenon-Coulombpartofthemolecularpotential26
where f ^ k , ^ r isthetransitionamplitudeand Z ¼ 1foraninitialneutral target.
OnequantitythatismeasurableexperimentallyisthePIcrosssection, definedtheoreticallyas
σ dE ¼ π e 2 m2 ℏ2 c ω g ðÞ Ψ0 b D g ðÞ Ψ DE 2 , (4) where ω L ðÞ ¼ E E0 or ω V ðÞ ¼ E E0 ðÞ 1 and c isthespeedoflight.
see Section4.2.1),configurationinteraction(CI,see Section4.1),ground stateinversionmethod(GIPM/D,see Section4.7.2),random-phase approximation(RPA,see Section4.9)andlogarithmicderivativeKohn method(LDKM,see Section4.11.1).Theyarefurthercomparedwith theexperimentaldataofChungetal.32 Forthisexample,SCFandCI calculationsusedOCE,andLDKMtheFCapproximation.Exceptfor theSCFresults,weseeanexcellentagreementbetweenalltheorieswith experimentaldata.Indeed,themoleculeH2 issufficientlysimpletoallow foraPIstudytakingintoaccountallinteractions.Oneaspect,though,that remainschallengingistocalculatepreciselythepositionsandwidthsofthe doublyexcitedstatesthatdependonthenuclearmotion.
Weshowin Fig.2 thePIcrosssectionsfortheoutervalenceorbital3σ g of N2.ForsuchMO,weshowcalculationsperformedwithCI(Section4.1), time-dependentdensityfunctionaltheory(TD-DFT,see Section4.3.2), multiple-scatteringXα (MSXα,see Section4.6),Stieltjes–Tchebycheff technique(STT,see Section4.10)anditerative-Schwingermethod (ISM,see Section4.12.1).NotethattheresultsforCIandTD-DFTwere obtainedusingOCE,andforISMusingtheFCapproximation.Thetheoretical crosssectionsarecomparedwiththeexperimentalresultsofPlummeretal.38
Figure2 PartialPIcrosssectioninMbversusphotonenergyineVfromtheMO3σ g of N2.ResultsforCI39 (red(darkgrayintheprintversion),dash);TD-DFT40 (green(grayin theprintversion),dash-dot);MSXα 41 (blue(darkgrayintheprintversion),dots);STT42 (gray,dash-dot-dot),andISM43 (orange(lightgrayintheprintversion),dash-dash-dot) arecomparedwithexperimentaldata38 (blackdots).
ThesituationchangesdrasticallywhenmovingfromH2 toamorecomplexmoleculesuchasN2.Thefiguresshowthattheagreementbetween differenttheoriesandexperimentaldataisbasicallylost,especiallyforenergiesclosetothethreshold.Moreover,onlyapartialagreementforhigher energiesisobserved.ExceptfortheCIresults,noneoftheothercalculations reproducesthedifferentseriesofresonanceslocatedbetween20and25eV.
ThePIcrosssectionsforCO2 areshownin Fig.3 fortheMO1π g.WecomparetheresultsobtainedwithGIPM/D(Section4.7.2),STT(Section4.10), ISM (Section 4.12.1)and R-matrixmethod (RMM,see Section4.8).The experimentaldata aretakenfromBrionandTan.44
Here,resultsforISMandRMMusedboththeFCandtheFNapproximations.Dependingontheenergyrange,thedifferenttheoreticalcalculationspresentagainonlyapartialagreement,andeveniftheycannot reproducecompletelytheexperimentaldata,theyperformratherwell beyond25eV.AlthoughthecenterofmassofCO2 isclosetothe Catombecauseofitslineargeometry,thismoleculeisparticularlydifficult todescribe:thedensityofchargeiscompletelydelocalizedaroundthemoleculeandonlytheuseofmulticenterwavefunctionsyieldsacceptablePI results,asintheGIPM/Dcase.
GIPM/D STT ISM RMM Experimental
Figure3 PartialPIcrosssectioninMbversusphotonenergyineVfromtheMO1π g of CO2.ResultsforGIPM/D45 (brown(darkgrayintheprintversion),dots);STT42 (gray, dash-dot-dot);ISM46 (orange(lightgrayintheprintversion),dash-dash-dot),and RMM47 (blue(darkgrayintheprintversion),solid)arecomparedwithexperimental data44 (blackdots).
TheMCTDHFhasbeenusedbyKatoandKono76 andbyHaxton etal.77 tostudyPIofH2 byintenselaserfields,andalsobyHaxtonetal.78 forHF.
Densityfunctionaltheory(DFT)iswi delyusedinquantumchemistry.It allowsforeasydeterminationoftheelectronicstructureforgivensystems (anatom,amolecule,acrystal,etc.),regardlessofitsextensionorthe numberofparticlesthatconstituteit.While“standard”quantum mechanicsworksdirectlywiththemany-bodywavefunctionsofthedifferentparticlesinagivensystem,DFTusestheone-electronelectronic density n r ðÞ andisbasedontwotheorems,calledtheHohenberg–Kohn theorems. 79.IndifferentimplementationsoftheDFTtostudyPIof molecules, n r ðÞ iscalculatedusingaconventionallinearcombination ofAOs(LCAO).49
4.3.1Kohn–ShamDFT
IntheKohn–ShamDFT(KSDFT),80 theHamiltonianofthemolecularsystemisdeterminedbythedensityoftheoccupiedorbitalsinthegroundstate andintermsoftheHartreepotential,theelectron–nucleiinteraction,and theso-calledexchange-correlationpotentialwhichcontainsallthe “unknowns”ofthesystem.Differentpotentialsareavailableintheliterature fordifferentatomicandmolecularsystems(see,forinstanceRefs. 81 and 82),based,forexample,onthelocaldensityapproximationoronthegeneralized gradient approximation.
TheKSDFThasbeenusedbyVenutietal.49 tostudyPIinC6H6;by StenerandDecleva,usingtheOCEapproximation,tostudyHF,HCl, H2O,H2S,NH3,andPH3 (Ref. 83),andCH4,SiH4,BH3,andAlH3 (Ref. 84).Toffolietal.,85 usingthemulticenterexpansion,calculatedcross sectionsforCl2,(CO)2,andCr(CO)6.WoonandPark86 alsostudiedC6H6 (benzene),C10H8 (naphthalene),C14H10 (anthracene)andC16H10 (pyrene).Strangesetal.87 studiedthedynamicsincirculardichroismof theC3H6O(methyl-oxirane).Toffolietal.88 studiedthePIdynamicsin C4H4N2O2 (uracil).
Time-dependentDFT(TD-DFT)89 constitutesanotherlineofdevelopmentoftheDFTmethods.Inthefirstordertime-dependentperturbative scheme,wherethezerothorderisequivalenttotheKSDFT,90 thelinear
responseoftheelectronicdensity n r ðÞ toanexternalweaktime-dependent electromagneticfieldcanbedescribedbyaSCFpotential,givenbyZangwill andSoven.91
TheTD-DFThasbeenusedbyLevineandSoven40 tocalculatephotoemissioncrosssectionsandasymmetryparametersofN2 andC2H2.Stener, Declevaandcoworkers,usingB-splines57 andtheOCE,studiedPIfordifferentmolecules:StenerandDecleva90 calculatedthecrosssectionsforN2 andPH3;Steneretal.92 forCH4,NH3,H2O,andHF;Steneretal.93 forCO andalsofromthe K-shell94;Fronzonietal.95 forC2H2;Steneretal.50 for CS2 andC6H6;Toffolietal.96 andPatanenetal.97 forCF4,andHolland etal.98 forpyrimidineandpyrazine.WealsofindtheworkofRussakoff etal.99 forC2H2 andC2H4,andbyMadjetetal.100 forC60.Differentresults formolecularPIhavebeenreviewedbySteneretal.101
Forthesakeofcompleteness,wealsomentionsomestudiesofmolecular PIthatuseaslightlydifferentapproach,thestatic-exchangeDFT:Ple ´ siat etal.102 investigatedPIofN2 andCO,andKukketal.103 fromthe inner-shellsofCO.
4.4.1ComplexScaling
Thecomplexscaling(CS)method104–105 hasbeenusedextensivelytostudy ionizationand,mainly,resonancephenomenainatomsandmolecules. Theideabehindthismethodistoscalethecoordinatesofallparticlesin theHamiltonianbyacomplex-valuedscalefactor: r ! re iθ .Onevariant oftheCSistheso-calledexteriorcomplexscaling(ECS),106–108 whereby thecoordinatesscaleonlyoutsideafixedradius R0
TheECSmethodhasbeenappliedtostudygeneralscatteringproblemsusing L2 basissetrepresentations.Itisespeciallywellsuitedtostudyionization processesinmolecules,sincethedefinitionoftheexteriorscaling (7) avoids complicatedscaling expressionsinthenuclearattractiontermsofthe Hamiltonian107 when R0 islargeenoughtoencloseallthemolecularnuclei.
TheECShasbeenusedmainlybyMcCurdy,Rescigno,Martı´nand coworkerstostudydifferentionizationprocessesinatomsandmolecules: McCurdyandRescigno109–110 usedCartesianGaussian-typeorbitals (CGTOs)tocalculatePIcrosssectionsofH + 2 ;Vanrooseetal.,111–112 using
SiF4,andSiCl4;PowisstudiedPIinPF3136,CH3I137,andCF3Cl.138 Finally, JurgensenandCavell139 compareddirectlyexperimentalresultswiththeMS Xα forNF3 andPF3.
4.7.1Plane-WaveandOrthogonalizedPlane-WaveApproximations
Thesimplestdescriptionofanionizedelectronistheplane-waveapproximation(PWA),butitisnotexpectedtogiveaccurateresultsnear threshold.140 Toourknowledge,thefirstimplementationsofthePWA areduetoKaplanandMarkin,141–142 LohrandRobin,143 andtoThiel andSchweig.144–145
Thefinalstateofthemoleculedescribesoneelectronthathasbeen excitedfromagiveninitialMOtoacontinuumnormalizedplane-wave orbital.140 Thisplane-waveisnotnecessarilyorthogonaltoanyoftheoccupiedMOs;iforthonormalityisimposed,wehavetheorthogonalizedPWA. ThePWAandtheorthogonalizedPWA,togetherwithSlater-type orbitals(STOs)todescribeAO,havebeenusedbyRabalaisetal.146 and byDewaretal.147 tocalculatePIcrosssectionsforH2,CH4,N2,CO, H2O,H2S,andH2CCH2.Huangetal.148 usedtheorthogonalizedPWA tocalculateangularasymmetryparametersforH2,N2,andCH4.Beerlage andFeil149 calculatedcrosssectionsforHF,(CN)2,CaHCN,C2(CN)2,N2, CO,H2O,furan,pyroleandtetrafluoro-pyrimidine.SchweigandThiel150 calculatedtherelativebandintensityofN2,CO,H2O,H2S,NH3,PH3, CH4,(CH3)2S,C6F6,amongothers.Hiltonetal.151 haveusedthe so-calledeffectivePWAtocalculatecrosssectionsforH2,CO,H2O,and C2H4.Finally,Deleuzeetal.152 usedtheorthogonalizedPWA,together withamany-bodyGreen’sfunctionframework,tocalculatePIcrosssections forCH4,H2O,C2H2,N2,andCO.
Theso-calledgroundstateinversionpotentialmethod(GIPM)hasbeen developedbyHilton,Hush,Nordholmandcoworkers151,153 withtheaim ofobtainingachemicaltheoryofPIintensities.154 Thismethodusesthestandardone-electronPWA,theorthogonalizedPWAortheenergyshifted PWA151 inordertocalculatetheelectroniccontinuumfinalwavefunction. Thecrosssectionisobtainedfromanatomicsummationtheorytogether withaplanewaveanalysisofdiffractioneffectsfromphotoelectronamplitudesfromdifferentatomsthatinterferewitheachother.35,154 ThemaindifferenceofGIPMwithastandardPWAisthatthepotentialfeltbyanelectron
whenleavinganatomiccenterinamoleculeiscalculateddirectlybyinversionofthegroundstateHForbital.153–154 TheGIPMtheorycaninclude threeimportanteffects:changeintheatomicorbitalsnatureuponformation ofthemolecule,diffractioneffects154 andexchangeinanexactsense.
TheGIPMhasbeenusedbyHiltonetal.tocalculatePIcrosssections forH2O155 andforH2,N2,andCO.35 AlsoKilcoyneetal.calculatedcross sectionsforH2,HF,andN2154;H2O,NH3,andCH4156;CO,CO2,and N2O,45 andforC2H4 andC6H6 51
Originallyintroducedinnuclearphysics,the R-matrixmethod(RMM)has beenadaptedtoatomicandmolecularphysicsbyBurkeandcoworkers(see Ref. 157 andreferencestherein).Applicationsofthismethod,inparticular for electron collisions,havebeenreviewedelsewhere.158–160 Theidea behindtheRMMistoenclosethescatteringparticlesandthetargetwithin asphereofradius a,sothatitshouldbepossibletocharacterizethesystem usingtheeigenenergiesandtheeigenstatescomputedwithinthesphere. Thenbymatchingthemtotheknownasymptoticsolutions,onecanextract allthescatteringparameters.The R-matrixisdefinedasthematrixthatconnectsthetworegionsinwhichthespaceisdividedinto.Theyare:(1)an internalregion,wherealltheparticlesareclosetooneanother,sothat theshort-rangeinteractionsandexchangeareimportant;(2)anexternal region,whereallparticlesarestillinteracting,buttheforcesaredirect andcouldhaveamultipolarcharacter.Inthemostconventionaluseof theRMM,theHamiltonianoftheinternalregionisdiagonalizedinorder toobtainthe R-matrixeigenenergiesandeigenfunctions,generallyusingthe nonadiabaticformalism.161 Theinitialandfinalstatesareexpandedinterms oftheseeigenstates.Thecorrespondingcoefficientsfortheinitialstateare usuallyobtainedbyperforminganall-channels-closedscatteringcalculation, andinthiscasetheproblemisreducedtofindthezerosofadeterminant.162–163 Toobtainthecoefficientsforthefinalstate,calculationsofelectronscatteringbythecorrespondingmoleculecanbemadeandtheresulting R matricesrepresenttheresultofafullnonadiabatictreatmentoftheinternal regionofthescatteringproblem,160 andprovidesthesolutionintheexternal region.164 Finally,withbothsetsofcoefficients,itispossibletocalculatethe requiredtransitiondipolemoments,andthusthePIcrosssection (4)
Since the correspondingformalismisrelativelynew,theRMMhasnot beenusedformoleculesasmuchasforatoms.However,wehavetheworks