Bound‐freespectrafordiatomicmolecules

DavidW.Schwenke

NASAAmesResearchCenter

M.S.T27B‐1,P.O.Box1000

MoffettField,CA94035‐0001

Itisnowrecognizedthatpredictionofradiativeheatingofenteringspacecraft

requiresexplicittreatmentoftheradiationfieldfromtheinfrared(IR)tothe

vacuumultraviolet(VUV).Whileatlowtemperaturesandlongerwavelengths,

molecularradiationiswelldescribedbybound‐boundtransitions,intheshort

wavelength,hightemperatureregime,bound‐freetransitionscanplayanimportant

role.Inthisworkwedescribefirstprinciplescalculationswehavecarriedoutfor

bound‐boundandbound‐freetransitionsinN2,O2,C2,CO,CN,NO,andN2+.

Comparedtobound‐boundtransitions,bound‐freetransitionshaveseveral

particularitiesthatmakethemdifferenttodealwith.Theseincludemore

complicatedlineshapesandadependenceofemissionintensityonbothboundstate

diatomicandatomicconcentrations..Thesewillbediscussedindetailbelow.

Thegeneralprocedureweusedwasthesameforallspecies.Thefirststepisto

generatepotentialenergycurves,transitionmoments,andcouplingmatrixelements

bycarryingoutabinitioelectronicstructurecalculations.Thesecalculationsare

expensive,andthusapproximationsneedtobemadeinordertomakethe

calculationstractable.Theonlypracticalmethodwehavetocarryoutthese

calculationsistheinternallycontractedmulti‐referenceconfigurationinteraction

(icMRCI)methodasimplementedintheprogramsuiteMolpro.1Thisisawidely

usedmethodforthesekindsofcalculations,andiscapableofgeneratingvery

accurateresults.Withthismethod,wemustfirstofchoosewhichelectronsto

correlate,theone‐electronbasistouse,andthenhowtogeneratethemolecular

orbitals.

https://ntrs.nasa.gov/search.jsp?R=20120015908 2020-06-18T23:32:31+00:00Z

Inallcalculations,weonlycorrelatethevalenceelectrons,i.e.the1s‐likeorbitalsare

keptdoublyoccupiedinallconfigurations.Weinitiallyconsideredcorrelatingall

electrons,whichwouldyieldmoreaccurateresults,howevertheextraexpensewas

deemedun‐warranted,giventhedifficultieswiththeicMRCImethodforhighly

excitedstates.Thiswillbediscussedinmoredetaillater.

Forallcalculations,theone‐electronbasissetusedwasbasedontheDunning2cc‐

pVTZbasisset,whichisthesmallestone‐electronbasissetcapableofyielding

reasonableresults.WealsousedthesecondorderDouglas‐Kroll‐Hess3

approximationtotreatscalarspecialrelativity.Tohelpdescribetheouter‐most

electronclouds,thecc‐pVTZbasissetwasaugmentedbyoneextradiffusefunction

foreachshellforCandN,andtwoextradiffusefunctionsforeachshellforO.The

extrashellsonOarerequiredduetotheimportanceofO‐configurations.4

Furthermore,systemsexpectedtohavelow‐lyingRydbergstates(theneutrals,but

notthecations)hadanadditionaldiffusefunctionaddedforeachshell.

Themolecularorbitalsweregeneratedusingthefollowingcompositeprocedure.

Wefirstgeneratedvalancemolecularorbitalsbycarryingoutaseriesofcomplete

activespace(CAS)calculationsstartingatlargeinternuclearseparations(100aofor

neutralsand1000aoforions)andworkingourwayintosmallinter‐nuclear

separations.Wewillusethesymbolrforinter‐nuclearseparation.About50

internuclearseparationswereconsidered.Inthesecalculations,theO2s‐like

orbitalswerekeptdoublyoccupiedinallconfigurations.IntheCAScalculations,

severalrootsfromeachsymmetrywereoptimized,withtheweightforeachroot

computedfromtheenergydifferenceasdescribedbyDeskevichetal.5Webasedour

selectionofrootsontheatomicstatesthatwecouldreliableconvergeto.Thusour

orbitalstransformasC∞vforheternucleardiatomicsandD∞hforhomonuclear

diatomics,andfurthermoreleadtotheproperatomicdegeneraciesinthe

asymptoticregion.Wejustconsideredasingleoverallelectronspin:doubletsfor

CN,N2+,CO+,tripletsforN2,O2,C2,CO,andquartetsforNO.ForN2,C2,COandNO.

Thischoiceofspinisnotthesameasthegroundelectronicstate,butthischoice

givesbettercoverageofpossibleatomicstates,thusitgivesbetteroverallresultsfor

allspinsintheicMRCIcalculations.Theresultsfromoneinternuclearseparation

wereusedasinitialguessforthenextsmallerinternuclearseparation,andthediab

procedureinMolprowasusedtomakethemolecularorbitals,bothactiveand

virtual,lookassimilaraspossibleastheinternuclearseparationchanged.

Oncethevalencemolecularorbitalsweredetermined,wedeterminedlow‐lying

Rydbergorbitalsfortheneutrals.Forthesecalculationswestartedatthesmallest

inter‐nuclearseparationsandworkedourwaytolargerrvalues.Thevalence

molecularorbitalswerekeptfrozenattheirpreviouslydeterminedvalues,andin

theCAScalculationswerestrictedtheconfigurationssothatexactlyoneelectron

waskeptintheRydbergorbitalofinterest.ForN2,O2,C2,andCO,weusedthe

lowestlyingRydbergsingletstates,whileforCNandNOweusedthelowestlying

Rydbergdoubletstates.Inasequentialmanner,wegeneratedtwoRydberg

orbitalsandbothcomponentsoftheRydbergorbital.Aswiththevalenceorbitals,

weusedthediabprocedureinMolprotoensurethattheRydbergorbitalssmoothly

changedwithinter‐nucleardistance.

Nowweturntoevaluatingtheenergiesandcouplingmatrixelements.Wecarried

outicMRCIcalculationsforusinganactivespacethatconsistedofallvalence

orbitalsandallRydbergorbitals:howeverO2svalenceorbitalswherekeptdoubly

occupiedinthereferenceconfigurationsandamaximumofoneelectronwas

allowedontheRydbergorbitals.Forthesecalculations,weusedamodifiedversion

ofMolprothatallowedtheselectionofforthestateofinterest.Thus,stateswere

computedfromseparateicMRCIcalculationsthenstates,etc.Nonetheless,itis

necessarytoextractmultiplerootsfromtheicMRCIcalculations.Thesecalculations

canbecarriedoutintwoways:thecoupledstatemethod,inwhichallrootsare

computedtogetherandTheprojectedstatemethod,inwhichallrootsarecomputed

sequentially.ThefirstapproachisthedefaultprocedureinMolproandhasthe

advantagesthatthecomputedwavefunctionsareorthogonalandhavelower

energies,andhencearemoreaccurate,thantheprojectedstateenergies.Ithasthe

disadvantagethatthecostofcalculationgrowssignificantlyasthenumberofstates

increasesMuchmoretroublesome,however,isthefactthatchangesinthe

characterofhigherstatescanleadtodiscontinuouschangesinlowerstateenergies.

Theadvantageoftheprojectedstatecalculationsisthatthecostislinearinthe

numberofroots,andstateenergiesarecontinuousastheinternuclearseparation

changes.Thedisadvantagesoftheprojectedstatecalculationsaretheelectronic

wavefunctionsarenolongerorthogonaland,ifoneisunlucky,astateflippingcan

occurthatcausesthecalculationstofail.Thusneithermethodwillalwaysgive

reliableresults.Forthemostpart,weusetheprojectedstatemethod,withthe

coupledstatemethodusedselectivelyforO2andCO+.Theunpleasantfeaturesthat

occasionallyoccurinhighlyingrootsinthefiguresshownismainlyduetothis

problem.

Usingtheseelectronicwavefunctions,wecomputedtheenergies,theoverlaps,the

dipolemoment,thequadrapolemoment,themagneticdipolemoment,andspin‐

orbitmatrixelementsforallpossiblecouplings.Furthermore,toascertainthe

relativephasesofmatrixelementsandtotransformfromtheadiabatictoadiabatic

basis,wecomputedtheoverlapsofthewavefunctionsfromadjacentinter‐nuclear

separations.

Nowletmakeaslightdigression.Theenergiescomputedasdescribedaboveare

knownasadiabaticenergies,forthenucleiare"clamped"tofixedpositionsinthe

calculationsoftheelectronicenergies.However,foraccuratecalculations,itis

importanttoconsidertheeffectofunclampingthenucleiontheelectronic

properties.Thisismostreadilyshowninthe2manifoldofNO:inthefigurethe

symbolsaretheadiabaticenergies,andthelinesareMoleculardiabaticenergies.

TheadiabaticenergiesshownumerousavoidedcrossingsastheRydbergstate

comesdowninenergy.Atthesecrossings,theelectronicwavefunctionchanges

rapidlyfromonesidetotheother,andthusthecouplingbetweentheinternuclear

coordinateandtheelectronicwavefunctionswillbeverylarge.Eventhoughthis

couplingcanbeincludedinthecalculationsinafairlystraightforwardmanner,

othercomplicationswithusingtheadiabaticelectronicfunctionsremain.Namely

thematrixelementsofotheroperators,suchasthedipolemoment,havevery

complexbehaviorinthevicinityofanavoidedcrossing,whichmakesreliable

interpolationdifficult.Itisthusadvantageoustotransformtoadifferentelectronic

basisthatdoesnothavetheseproblemsbecausethecharacteroftheelectronic

wavefunctionshasbeenforcedtovaryslowlyandsmoothly.Thistypeofbasisis

calledadiabaticbasis.Severaldifferentchoicesforthediabaticbasisarepossible,

andtheparticularbasisweusewecalltheMoleculardiabaticbasis.Thisdiabatic

basisisformedrecursivelyrequiringthattheadiabaticanddiabaticbasisbethe

sameattheinter‐nuclearseparationwheretheadiabaticpotentialenergycurvehas

itsdeepestminimum.Inpractice,westartwiththeinternuclearseparationatthe

lowestminimum,thenpropagatebothforwardandbackwardsinr.Thepropagation

involvesaseriesofJacobi‐typerotations,appliedtotheoverlapbetweenthe

electronicfunctionsatneighboringgeometriestomakeitasdiagonalaspossible.

TheJacobirotationsareonlyappliedtoonesideoftheoverlapmatrix,andeach

individualrotationangleischosentominimizetheoffdiagonaloverlapofeachpair

ofstates.Sweepsthroughthematrixarecarriedoutuntiltheoffdiagonalsofthe

matrixstopdecreasing.Inthediabaticbasis,off‐diagonalcouplingduetoelectronic

energyisnon‐zero,butallcurvesaresmoothandeasytointerpolate.The

disadvantageofthisbasisisthatconvergencetotheadiabaticatomicenergiesis

ratherslowintheasymptoticregion.Howeverthisisnotexpectedtobeacritical

partofpredictingaccuratemolecularspectra.

Anotherthingtonoteaboutthisfigureisthediabatizationprocedurenotonlyturns

thesharplyavoidedcrossingsintorealcrossings,butalsochangesthelarger‐

dependenceofthegroundstatepotential.Inthelargerregion,thegroundstate

potentialdoesnotexhibitanyobviousavoidedcrossings,sothatthediabaticcurve

differsfromtheadiabaticcurveinthisregionmightbesurprising.Howeversome

considerationofthesituationmakesthesituationmorereasonable.Inthelarger

region,thepotentialenergycurvescanbelabeledbyatomicquantumnumbers,e.g.

thelowestadiabaticpotentialcurvehastheasymptoteof3PO+4SN.Inthevicinity

oftheminimum,nearr=2ao,thewavefunctionwillreflectamixtureofatomic

states.Forexample,duetothehigherelectro‐negativityofOoverN,therewillbe

significantO‐N+characterinthewavefunction.IntheMoleculardiabaticstates,the

fractionalcharacterofthewavefunctionismoreorlessfrozenatitsvalueatthe

minimum,thustheenergyattheasymptotewillbelargerthanthelowestadiabatic

curve.Thisisexactlywhatwesee.Whathappentothe3PO+4SNstateinthe

moleculardiabaticbasis?Afterall,eveninthevicinityoftheminimum,themolecule

isnotpurelyO‐N+.TheO+Ncomponentshouldstillbepresent.TheNOfigure

appearstoshowadeficiencyinthemoleculardiabaticbasis.Thelossofthe3PO+4S

Nstatewouldmeanthatonecannotusethisbasisfordissociation.

Itisinterestingtocomparethecomputedpotentialenergycurveswithexperimental

data.Inthefigures,wehaveplacedlabelsofelectronicstatesatthevalueofTeand

retakenfromHuberandHerzberg.6Forthemostpart,theagreementisextremely

good,graphically.ThetableabovewiththecomparisonsbetweenthecomputedTe

andthosegivenbyHuberandHerzberginwavenumbers.Whilethedifferencesare

asmallfractionofTe,theabsoluteerrorsaresizableenoughtobeobservedinsaya

spectrumfromtheEASTfacility.Thusweshiftedthecurvesforwhichdatawas

availablebythenegativeoftheamountlistedinthecalc‐obscolumns.Amore

accurateprocedurewouldbetouseT00,whichisamoredirectlymeasuredquantity,

buttheotherapproximationsinthepresentworkmakethislevelofrefinement

superfluous.

Nowitshouldbenotedthatthiscomparisonandcorrectionisnotexact,forthe

valuesofTecomputedarejustthedifferencesinminimumenergiesintheMolecular

diabaticcurves,whiletheTefromHuberandHerzbergiseithertheY00parameterin

theDunhamexpansionoftheexperimentalenergylevelsforsimplecases,ora

parameterinacomplicatedeffectiveHamiltonianformorecomplexcases.Thuswe

arecomparingapplestooranges.Nonethelessitisexpectedthatthiscorrectionwill

bereliableenoughtoyieldusefulresults.

Wenextcarriedoutcoupledro‐vibrational‐spin‐electroniclevelcalculations.To

proceedwebuildupbasisfunctionsforallcoordinatesexcepttheinter‐nuclear

separation:

with , and0when=0.Here isthespin‐electronicfunction,

withlabelingparticularelectronicstates,Sisthetotalelectronicspinquantum

number,istheabsolutevalueofquantumnumberspecifyingtheprojectionofthe

totalelectronicangularmomentumontheinter‐nuclearbondaxis,isthequantum

numberspecifyingtheprojectionofthetotalelectronicspinangularmomentumon

theinter‐nuclearbondaxis,and specifyspin‐electroniccoordinatesintheframe

ofreferencehavingthebodyfixedzaxisalongtheinter‐nuclearbond,JandMare

thequantumnumbersforthetotalangularmomentumandtheprojectionofthe

totalangularmomentumonthespacefixedzaxis.Thebodyfixedandspacefixed

axisarerelatedbytherotationbytheEulerangles0,Pistheparityand=Ne/2,

whereNeisthenumberofelectrons.Thefactoris‐1for‐states,and+1

otherwise.TheW aretheWignerrotationmatrixelementsasdefinedbyEdmonds.7

Inthisbasis,thefullrelativisticHamiltonianoperatorisdiagonalinJMP.Thenon‐

relativisticpartoftheHamiltonianisalsodiagonalinS.Ifalloffdiagonalcoupling

matrixelementsareneglected,thenthiswouldbewhatiscalledtheHund'scasea

basis.However,wedonotmakethisapproximation,andratherincludeallcoupling

terms.ThuswecanaccuratelytreatallpossibleHund'scasesaswellasall

intermediatecases.

Thematrixelementsofthenuclearrotationalangularmomentumcontribution

R(R+1)tothekineticenergyisgivenby

S

JMP 2J 1

8 10 0

12

S ˜ x W M(J ) 0 P 1 J S ˜ x W M

(J ) 0

P 1 0 S ˜ x

˜ x

Inthisexpression,weonlyneglectthe term,foritiscurrentlydifficultto

calculate,andfurthermoreonlyvariesbyafewcm‐1acrossthepotentialenergy

curve.

Besidesthecustomaryelectricdipoletransitionmoment,wewillconsiderhigher

ordermoments.ThisisbecausetheLBHaXN2transitionisdipoleforbidden.The

nextordermomentsaretheelectricquadrapolemomentandthemagneticdipole

moment.Ingeneral,thetransitionmatrixelementsinthisbasisaregivenby

JMSP R(R 1) J M S P J J M M S S P P

J J 1 S S 1 2 2 Lx2 Ly

2 1 J J 1 S S 1 10 0 0

12

12 Lx

J J 1 1 0 0 0 12

1 S S 1 1 0 0 0 0 12

P 1 J 0 0 J J 1

12

1, S S 1 12

12P 1 J 1

2 0 J 1

2 S 12

1 0 0 Lx J 12

Lx2 Ly

2

where isa3‐jsymbol.NowthisfactorizesintoatermthatcontainsalltheM

dependenceandtherest:

Nowourfinalwave‐functionswillconsistoflinearcombinationsofthe with

fixedJMP,butthiswillnoteffectthisfactorization,butwillchangefromthelabels

to.Thentherateofstimulatedemission(akaEinsteinAcoefficient)from

electricdipoleradiationis

therateofstimulatedemissionfrommagneticdipoleradiationis

JMPS ˆ O mqt

m

W mm '(q ) J M P S 1 M 2J 1 2 J 1

10 0 1' 0 ' 0

12

S S

J q J

M m M

1 P P c t 1 q

2

J q J

Sqt S

0P' ' 1 J ' J q J '

' '

Sqt ''S '

. . .

. . .

JMPS ˆ O m

qt

m

W m m (q ) J M P S 1 M J q J

M m M

FS S

JP J P qt

SJMP

S

andtherateofstimulatedemissionfromelectricquadrapoleradiationis

Weseethefollowingselectionrules:forelectricdipoleradiation(e1), and

,whileforbothmagneticdipole(m1)andelectricquadrapole(e2)

radiationwehave ,andadditionallyfore2, ispossible.Withall

otherthingsbeingapproximatelyequalinatomicunits,m1isdownfromdipole

allowedtransitionsbyafactorof1/c2,orabout10‐4.Comparedtom1,e2isdown

aboutby2,whichaboutanother10‐4intheinfra‐redregion,butnearingunityin

theVUV.

Sofarwehavejustdiscussedourtreatmentoftheangular‐electronicdegreesof

freedom.Theradialdegreeoffreedom,r,istreatedusingthefinitedifference

boundaryvaluemethodincludingfullcoupling.Weperformcalculationsofeach

valueofJandP(theradialwavefunctionsareindependentoftheMquantum

number)andweexplicitlyintegratethetransitionmomentsovertheradial

functions,thuswedonoteneedHönl‐Londonfactorsorther‐centroid

approximation.Furthermore,byfullyincludingallcouplingterms,weexplicitly

obtainnonzerolambdadoublingsplittings,spin‐forbiddentransitions,andall

perturbations.

Whensolvingfortheradialfunctions,wehavetwosituationstoconsider.Inthe

first,forboundlevels,theboundaryconditionsarehomogenousas andas

.Inthesecond,thetotalenergyisabovethelowestdissociationlimit,thusthe

largerboundaryconditionsareinhomogenous.Thiscomplicatesthings

considerably.

P P

J J 1

P P J J 2

Forboundlevels,thehomogeneousboundaryconditionsmakethesolutionofthe

radialfunctionsabandedeigen‐valueproblem.Thustheenergiesarefixedbythe

boundaryconditions.

Ontheotherhand,forun‐bound,orfreelevels,theenergyisafreeparameter,and

wemustemployscatteringtheorytodeterminethewavefunction.Thenlarger

boundaryconditionstaketheform

wherelabelschannels(channelsarethevariousasymptoticallydiagonalbasis

functions), ,whereisthereducedmassforradialmotion,Eisthe

kineticenergyoftherelativemotionofthefreeparticles,isthethresholdenergy

forchannel,andSisthescatteringmatrix,whichisacomplexsymmetricmatrix.

Fornotationalsimplicity,wehavesuppressedtheenergylabelfromtheradial

functionfandthescatteringmatrix.Nowwedonotknowapriorithevaluesofthe

scatteringmatrix,soinpracticalcalculationswesolveforthefunctions subject

totheboundaryconditions .Wethenanalyzethelargerbehaviorof

thenumericalresults,andformlinearcombinationstosatisfythedesiredboundary

conditions.Itshouldbenotedthattheaboveexpressionsareonlyvalidwhenkis

real,i.e.E>.Forotherchannels,theradialfunctionsdecayexponentiallyatlarger.

Nowforfreestates,howshouldwesampleE?Theobviousansweristousea

densityofenergiessothatthescatteringmatrixcanbereliablyinterpolatedover.

Thisrequiresknowledgeoftheformalbehaviorofthescatteringmatrix.Thereare

polesinthescatteringmatrixatdiscretecomplexenergies ,wheren

isacountingindex.Whilecomplexenergiesarenotphysicallyobtainable,these

poleshaveimportantconsequencesontherealenergyaxis.Specifically,atenergies

"far"from ,i.e.greaterthanaboutnaway,thephaseofthedeterminantofS

decreasesmonatonicallywithincreasingE,butnear ,thephaseabruptly

f r ~ k 1

2 exp ikr exp ikr S

k 2 E

˜ f

˜ f rbc

E Xn in /2

Xn

Xn

increasesby2like .Physically,thesepolescorrespondto

"almost"boundstateswithdecaylifetime/n.ThusweneedtoconcentrateourE

samplingnearthe .Butonceagain,wedonotaprioriknowthe .

Weobtainestimatesofthe byenclosingthesysteminafinitebox,whichturns

allfreestatesintoboundstates.Thenwecandetermineenergyeigenvalues,and

the willbenearbysomeoftheseeigenvalues.Byevaluatingtheenergy

derivativeofthephaseofthescatteringmatrix,wecanisolatethe ,andthendoa

refinementtodeterminebetterestimatesof andn.

Itwouldseemthatknowing andnandthe ,wewouldbeinapositionto

computebound‐freetransitionsjustlikebound‐boundtransitions,butfurther

reflectionrevealsadditionaldetailsthatmustbeaddressed.Firstofall,howshould

the be"normalized"?Thisisnotwelldetermined,sincetheyarenotsquare

integrablefunctions.Secondly,whichofthe'shouldwechosetocompute

transitionmatrixelements?Anotherwaytolookatthislastproblemis

representsincomingfluxinchannel'andoutgoingfluxinallopenchannels.But

thephysicalsituationshouldhavenoincomingflux,sinceallthepopulationwould

begeneratedbyphotonabsorption.

Thepropersolutionofthisproblemistodirectlycoupletheradiationfieldtothe

nuclearproblemusingGreen'sfunctions.Thisleadstoadrivingterminthe

differentialequationsfortheradialfunctions,andasolutionthatlookslike

withlindexingtheparticularboundstateforthetransition.Theamplitude is

complexanddependsonenergy.Notethatthesedifferentialequationsarenomore

difficulttosolvethanthescatteringequations.Thenonefindsthatif corresponds

2tan1 n /2 Xn E

Xn Xn

Xn

Xn

Xn

Xn

Xnf

f

f

gl r ~

2k

12

Ml exp ikr

Ml

Xn

Formatted: Lowered by 4 pt

Formatted: Lowered by 4 pt

toaverynearlyboundstate,theninthelimitofthecouplingofthetrueboundstate

tothecontinuumbecomeszero,thenifEisnear ,

.

Thatis,theemissivitylookslikeanEinsteinAmodulatedbyaLorentzian,andthat

theEinsteinAcanbecomputedvia

.

Howeverthissimpleresultisnottrueingeneral.Thisisbecausethe arecomplex

numbers,andthusvariouspolescancauseconstructiveanddestructive

interference.Seethefollowingfigureforanexample:

Inthisfigure,theredis ,whichclearlyshowsfirstderivativelikefeatures

ratherthanLorentzianfeatures.Thegreenistheresultofconvolutionwiththe

Dopplerbroadening.Oneseesthatthebroadeningforthesesharpfeaturescan

resultinmuchofthepositiveportionofthefeaturebeingeliminated.Inour

calculations,wefindthatthissortofsituationistheruleratherthantheexception.

Thusitisnotsufficienttojustevaluatetheamplitude atthe ,butratheritis

importanttohavemoreenergiesinordertonaildowntheenergydependenceof

theamplitudeinordertoperformtheDopplerbroadeningcalculation.These

Xn

M

l 2

An

l n

2E Xn 2

n2 /4 1

Anl M

l 2

E Xn

Ml

Ml 2

Ml

Xn

Formatted: Lowered by 4 pt

Formatted: Lowered by 4 pt

calculationsareinprogress,buthavenotyetbeencompleted.

Anadditionalcomplicationisthatinadditiontoemissionfromanalmostbound

state,therewillbecollision‐inducedemissionatthatenergy.Theemissionfromthe

almostboundstatewillbetheproductofthealmostboundstatepopulationtimes

.Incontrast,thecollision‐inducedemissionwillbetheproductofthe

atomicpopulationsforchannel'timesthefactor .Seebelowforan

example.Theredlinesareformolecularemissionwhiletheothercolorsarefor

collision‐inducedemission.

Thusweonlyconsiderbound‐boundtransitionsinthisdocument.Inthefollowing

figures,weshowtheopticallythinabsorptionspectraofthediatomicscomputedat

T=10,000K,whichisroughlyappropriateforEarthentry.

WeseethatinallcasesexceptforN2,thedipoletransitionsdominatethespectrum.

Wealsoseesignificantfillinginofthespectrabetweenpeaksduetoweak

transitions,includingspin‐forbiddentransitions.

Ml 2

Ml S

2

References:

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