line shape parameters for the H O-N collision system Vibrational dependence, temperature dependence, and prediction of Journal of Quantitative Spectroscopy & Radiative Transfer

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Journal of Quantitative Spectroscopy & Radiative Transfer

journalhomepage:www.elsevier.com/locate/jqsrt

Vibrational dependence, temperature dependence, and prediction of line shape parameters for the H ₂ O-N ₂ collision system

Bastien Vispoel

^a,b,c

, João H. Cavalcanti

^a

, Evan T. Paige

^a

, Robert R. Gamache

^a,∗

aDepartment of Environmental, Earth, and Atmospheric Sciences, University of Massachusetts Lowell, Lowell, MA, 01854 USA

bResearch Unit Lasers and Spectroscopies (LLS), Institute of Life, Earth and Environment (ILEE), University of Namur (UNamur), 61 rue de Bruxelles, B-50 0 0, Namur, Belgique

cRoyal Belgian Institute for Space Aeronomy (BIRA-IASB), 3 Avenue Circulaire, 1180, Brussels, Belgium

a rt i c l e i n f o

Article history:

Received 26 February 2020 Revised 11 April 2020 Accepted 12 April 2020 Available online 29 May 2020

a b s t r a c t

Inarecentwork[JQSRT228,79(2019)],Vispoeletal.optimizedtheintermolecularpotentialusedinthe ModifiedComplexRobert-Bonamy(MCRB)formalismfortheH₂O-N₂collisionsystem.Calculationswere madeforanumberoftransitionsintherotationandν2 bands.Theneedsofthespectroscopicandas- trophysicscommunitiesincludedataforwater vaportransitions thathavemultiple vibrationalquanta exchanged.Inthiswork,MCRBcalculationsweremadefor0–4vibrationalquantaexchangedintheν1, ν2,and ν3 bandsfor 13temperaturesfrom 200–3000K; 7272transitions foreach band.From these data,thevibrationalandtemperaturedependenceofthehalf-widthandlineshiftweredetermined.The temperaturedependencewasdeterminedusingtheGamache-Vispoelmodel[JQSRT217,440(2018)].The dataallowedthedevelopmentofaroutinethatcanpredictthehalf-width,lineshift,andtheirtempera- turedependencefortransitionsnotyetstudied.Thepredictionalgorithmisbasedontheory[JQSRT,83, 119(2004)]andyieldslineshapeparameterswithmuchsmalleruncertaintythanobtainedbyfittingwith adhocpolynomialsorJ” averagedvalues.Alinefilebasedonthe2020updatetotheHITRAN2016wa- tervaporlinefilewascreatedwithN₂asthebroadeningspecies.Thesedataareusefulforcombustion studiesandasafirststeptodeterminingair-broadeningfortheHITRANandGEISAdatabases.

© 2020PublishedbyElsevierLtd.

1. Introduction

Watervaporisseen throughouttheuniverse.It isamajorab- sorberofinfraredradiation,aswellasotherfrequencies.Inourso- larsystem, watervaporhasbeenmeasuredin thefourterrestrial planets [1–4],intheatmospheresandinteriorsofthefourJovian planets [5], andrecentlyonEuropa,a moonofJupiter [6].Water has been detected in comets [7], asteroids[8], andin the dwarf planet Ceres [9]. It has been seen in the atmospheres of brown dwarfsandcoolstars[10–15],whereitcandominatethespectrum asit doesinEarth’satmosphere. Waterhasbeendetected inthe atmospheresofanumberofexoplanets [16–32]andisone ofthe principalspeciesusedasahabitabilityindicator.

These studiesof watervapor haveutilizedremote sensingin- strumentstocollectdata.Thereductionofthemeasurementsinto usefuldatareliesonradiativetransfermodels,whichinturnneed high-quality spectroscopic parameters. In particular, the line po- sition, intensity, lower state energy, half-width, line shift, parti-

∗ Corresponding author.

E-mail address: [email protected] (R.R. Gamache).

tion sums andtheir dependence on temperature. For the atmo- sphereofEarth,muchofthesedatacanbe foundontheHITRAN [33]orGEISA[34]databases.Ofthespectroscopicparameters,the pressure-broadenedhalf-widthremainsthemajorsourceofuncer- tainty in the reduction ofremotely senseddata in Earth’satmo- sphere.Inthelowertroposphere,theuncertaintyinthehalf-width corresponds1:1withtheuncertaintyintheresultingprofiles[35–

37]. Itis observed thatsmall changesin thehalf-widthcan have largeeffectsontheuncertaintyoftheretrieveddata[38–40].

TheHITRAN andGEISAdatabasesbothcontain half-widthsfor air-andself-broadening,however,andHITRANnowcontainsdata forother broadening species. Data for the pressure induced line shifts andtemperature dependence of the half-widthare sparse.

Measurementscanbemadeusingeitherairasthebroadeninggas orusingnitrogenandthenoxygenasthebroadeninggasandthe airbroadenedlineshapeparameterscanbedeterminedassuming binarycollisionsandDalton’slawofpartialpressure

γ

air=0.79

γ

^N2+0.21

γ

^O2,

δ

air=0.79

δ

^N2+0.21

δ

^O2, (1) As such, there has been a good number of studies devoted to determining line shape parameters for H₂O-N₂. These mea- surements were summarized in several intercomparison studies https://doi.org/10.1016/j.jqsrt.2020.107030

0022-4073/© 2020 Published by Elsevier Ltd.

2 B. Vispoel, J.H. Cavalcanti and E.T. Paige et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 253 (2020) 107030

[41,42].Thedatabaseofmeasurementscontainsalmost8000half- widths, ~7000 line shifts, and a small number (76) of temper- ature exponents for the half-width. These studies showed, in general, that the measurements do not agree within the stated uncertainties.

Thesemeasurements arefartoofew tofilladatabase,e.g. HI- TRAN2016[33]contains402900transitionsforwatervapor.Thus, researchers must rely on theory to compute the line shape pa- rameters.WhileMCRB theoryis capableofcomputinglineshape parameters and their temperature dependence for all transitions of H₂O, a rough estimate of the computation time required to computeallro-vibrationaltransitionsoftheprincipalisotopologue, H216O,isjustoverayearandahalf.Thus,thereare fartoomany transitions even for theory to consider, so that methods to pre- dictorestimate thelineshapeparameters areneeded. Anumber ofmethods havebeenusedtoestimate half-widths;a numberof studies useda polynomial inJ andK (Ka for asymmetricrotors) [43–47].TheFamilyofTransitionsmethod[48–51]andthesmooth variationandpairidentityrulesofMaetal.[52]canprovidehalf- widthsandlineshifts. Thesethree methodsonly providedata in therangeofthecurrentdataasextrapolationdoesnotworkwell withpolynomialsandcannotbedonewiththeothermethods.The half-widthscanbe estimatedusingthe energyaveragedifference (AED)method[53,54]givingdatawithuncertaintiesfrom15–30%.

Finally,some haveused half-widthsaveraged asa function ofJ”;

asshownbelow,thismethodcomparedwithdatacanhaveuncer- taintyofhundredstomorethanathousandpercent.

Gamache and Hartmann [55] developed a method to predict half-widthsandlineshiftsthatisbasedonsemi-classicalComplex Robert-Bonamytheory [56]. Themethodwassuccessfullyapplied by Jacquemart et al. [57], however scatter in the measurement databaselimitedtheir results.Morerecently,Gamache etal.[58–

61]haveshownthat givena numberofaccurate lineshapemea- surements,including temperaturedependence,the intermolecular potential used in the theory can be adjusted allowing the com- putationofanumberofvibrationallydependenttransitions.From thesecalculations,predictionroutinesbasedontheorycanbede- veloped[55,57,62,63]andaccurate lineshape informationcan be providedtothedatabases.

Inapreviousstudy,Vispoeletal.[64],usingdatafromtheH₂O- N₂ measurementdatabase[42],optimizedtheH₂O-N₂intermolec- ularpotential.Inthiswork, thepotential ofVispoeletal.is used tomakecalculationsfora numberoftransitions in13vibrational bands.Thesedatawere usedtostudythevibrationaldependence ofthe H₂O-N₂ collision system and to develop a line shape pa- rameterpredictionroutine.Inaddition,asdescribedbelow,calcu- lationsweremadeforasmallnumberoftransitionsfor10combi- nationbandstotestthepredictionalgorithm.

2.0. Theoryanddata

The calculations of the half-width and line shift were made usingthecomplex implementationofthe Modifiedsemi-classical Complex Robert Bonamy theory [65] and is designated MCRB.

In this formalism the half-width,

γ

^{, and line shift,}

δ

^{, for a ro-}

vibrationaltransitionf←iaregivenbytherealandimaginaryparts oftheexpression

( γ

+i

δ )

f←i= n2

2

π

^c

_∞

0

v

f

( v ₎

dv

_∞

0

2

π

^b

1−e^{−iS1+Im(S2)J2}e^−Re(S2)J2

db (2)

wheren₂ isthenumberdensityoftheperturbers (collisionpart- nerofH₂O),cisthespeedoflight,vistherelativevelocity,f(v)is theMaxwell-Boltzmanndistribution ofvelocity, <>J2 isan aver- ageover thestates ofthe perturber, andb is the impactparam- eter. The first and second order terms in the successive expan- sionof the Liouville scatteringmatrix, S₁ andS₂, depend on the

ro-vibrational statesofthetransition, the associatedcollisionally- induced jumpsfromthesestates,on theintermolecularpotential, and the characteristics of the collision dynamics. The MCRB for- malismisasemi-classicalmethodtodeterminethelineshapepa- rameters. Semi-classical designates that the internal structure of thecollidingmoleculesistreatedviaquantummechanics andthe dynamicsofthecollisionprocessaredeterminedviaclassicalme- chanics. Thus, inan optical transitionfrom initial state i to final state f,the active molecule will undergocollisionswith thebath molecules, forwhich the trajectories of the molecules are deter- mined using Hamilton’s equations. In these collisions, the initial andfinal statesoftheradiatingmolecule, iandf,andperturbing molecule,J2,willundergocollisionallyinducedtransitionstostates i’,f’,J₂’interrupting theradiationandcausingcollisionalbroaden- ing.Thestatesi’andf’arecalledthecollisionallyconnectedstates of theradiator andthe states J₂’ are calledthe collisionallycon- nected statesof theperturber. These collisionallyinduced transi- tions are given by selection rules determined by the wavefunc- tionsandtheintermolecularpotential.Giventhelargenumberof termsintheintermolecularpotentialusedinthiswork,thereare manystates,i’orf’orJ₂’,thatarepossible.Thequantummechan- ical components ofthe calculation are theenergies ofthe states involvedinthecollisionallyinducedtransitionsandthewavefunc- tionsforthesestates,whichareusedtodeterminetheprobability ofacollisionallyinducedtransitionfortheradiatorandperturber molecules.This probability isgivenby a reducedmatrix element [61,66,67].

Data are needed to study the vibrational dependence and to develop a prediction routine. Each transition is defined by a set of vibrational and rotational quantum numbers. Given the rota- tionalquantumnumbers,thepredictionroutinecandetermineline shapedata foranyset ofvibrationalquantumnumbersmaking a ro-vibrationaltransition.Thusuniquerotationalquantumnumbers are needed forthe calculations, e.g. the 7 ₃₅ ← 8 ₄₄ rotational transitionisinboththe

ν

1 and

ν

2 bands(andmanyother bands) but7₃₅←8₄₄onlyneedstobeinthelistonce.Usingaprelim- inary version of the HITRAN2020 database, largely based on the experimentaldatafromHITRAN2016[33]andabinitiocalculations from Conway et al./ [68], all unique rotational transitions with J”≤23forthe 3principalisotopologueswereselected.Thisproce- dure generateda listof7272transitions.MCRB calculationswere made for thesetransitions for0–4 vibrational quanta exchanged inthe

ν

1,

ν

2 and

ν

3 bandsat13 temperaturesbetween200and 3000K.Thesedatawereusedtostudythevibrationaldependence ofthehalf-widthandlineshift.The datadisplayedan unphysical structureforcertaintransitions.Figure1showsthehalf-widthsin cm⁻¹ atm⁻¹ unitsversusthenumberofquantaexchanged inthe transitionsfor

ν

κ=0to 4for

ν

1,

ν

2,and

ν

3; therotation band is given by a magenta solid star symbol (

ν

κ=0), the

ν

1 se- quenceisgivenbytheblueopencirclesymbol,the

ν

2 sequence isgivenbytheredasterisksymbol,andthe

ν

3sequenceisgiven by theblack solid circlesymbol.The theoretical modelforvibra- tionaldependence[55]suggeststhatthesymmetricandasymmet- ricstretchvaluesforthesame

ν

^{shouldhavesimilarvaluesand}

varysmoothly.Inthefigure,thevaluesfor

ν

1=

ν

3=4differsig- nificantly. Thereare manyother plotsthat were madewherethe

stretchmodeswith

ν

κ equal3or4thevaluesdiffermorethan whattheorywouldsuggest.Aftersomeanalysis,itwasfoundthat thetermvaluesusedinthecalculationshaveproblems.Theterm valuesarefromMARVELanalysisforwatervapor[69]wheremiss- ing termvaluesare supplemented usingab initioorapproximate

Fig. 1. Half-widths in cm ^-1atm ^-1units determined using the augmented MARVEL term values versus the number of quanta exchanged in the transitions for νκ= 0 to 4 for ν1, ν2, and ν3; the rotation band is given by a magenta solid star symbol ( νκ= 0), the ν1sequence is given by the blue open circle symbol, the ν2se- quence is given by the red asterisk symbol, and the ν3sequence is given by the black solid circle symbol. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

term values [70,71]. Figure 2 shows three panels of termvalues fortheJ=10, 11,and12rotationalstatesforthe(400) vibrational state versus Ka. The + symbols are for states with Ka+Kc odd andthexsymbolsare forstateswithKa+Kc even;MARVELdata usethecolorblueforoddstatesandredfortheevenstates.The black open circlesymbolslabeldata missinginthe MARVELlist;

thesedataaretakenfromtheabinitiocalculationsofPartridgeand Schwenke [71] or using the approximation Evr=E_vib+Erot. These datadonotfollowtherulesstatedbyMa etal.[52].Wherepair- ingshouldoccur,therearelargedifferencesinthetermvalues.For example,atJ= 10,Ka=7or8thedifferenceinthetermvalues are 459and420 cm⁻¹, respectively.There are similar differences for J= 11 or12. Forother states studied, similar differencesare seen,withsomelargerdifferences.Thesedifferencesaffectthede- terminationofthecontributionfromtheresonancefunctionsthat istakenfromtheenergygapsbetweenstate itoi’inacollision- allyinducedtransition.Becausetheresonancefunctionsaresteep, energygap errorsof hundreds ofwavenumberscan resultinthe contribution from the resonance function being off by orders of magnitude.

Tocorrectthisproblem,amoreconsistentandcompletesetof term values is needed. An advantage of term values determined by ab initiocalculations isthat they arecomplete. The drawback is that inthe abinitio calculation onlyJand symmetryare good quantum numbers, thus more work is needed for a full attribu- tion, i.e. assigning the Ka and Kc values to each state. The data of Partridge andSchwenke [71], labeled VTET, satisfy these con- ditions quite well. There is some mislabeling inthe data set but thesewerecorrectedusingtherulesofMaetal.[52].Fig.3shows the VTET data for the same states of Fig. 2. The + symbols are forstates withKa+Kc oddandthe xsymbolsare forstateswith Ka+Kceven, VTETdatausethe colorblueforoddstatesandred fortheevenstates.

The MCRB calculations were repeated using the VTET term values. Fig. 4 shows the half-widths for the same transition as Fig. 1butdetermined usingthe VTETenergies. Symbolsandcol- ors are the same as Fig. 1. Note, the data are now more physi- cal andconfirm the theory.The data forother transitions follow the theory similarly, giving a data setthat can be used to study the vibrational dependence and be used to develop a prediction routine.

3. Vibrationaldependenceofthehalf-width,lineshift,and theirtemperaturedependence

Giventheupdateddata,vibrationaldependenceplotssimilarto Figs.1and4weremadeforallthetransitionsofthisstudy(7272).

Thelistoftransitionsisquitecompleteallowingconclusionstobe drawnwithconfidence.Figs.5and6,respectively,show examples wherethestretchmodedata(

ν

1 or

ν

3 sequence)are similarand whentheyaredifferent.TheplotsymbolsarethoseofFigs.1or4.

Mostoftheplotsshowsimilartrendsfor

ν

1 and

ν

3(asinFig.5), however, too many show differences between

ν

1 and

ν

3 (as in Fig. 6). Note, if all plots resembled Fig. 5, it would suggest that thesymmetricandasymmetricstretchdatacouldbemodelledto- gether.The factthat there are anumber oftransitions forwhich thesymmetricstretchandtheasymmetricstretchdataare differ- entimpliesbetterpredictionswillbeobtainediftheyaremodelled separately.Thisfactisdiscussedmorebelow.

Thevibrational dependenceofthe half-widthandlineshift is considered asa function of temperature. In Figs. 7a-7c, the half- widthandline shiftat sixtemperatures are plottedforthe

ν

1

(Fig.7a),

ν

2 (Fig.7b), and

ν

3 (Fig.7c)sequences forthe7_6.1

←6._4.2rotationaltransition.Shownarethedatafor200K(black asterisk symbol), 296 K (red open circle symbol), 700 K (green open square symbol), 1250 K (blue open triangle symbol), 2000 K(orangesolidcirclesymbol),and3000K(purplesolid starsym- bol).Fig. 7a and cshows a noticeable vibrationaldependence of the half-widthand lineshift forthe 200 K and296 K data; the 700Kdatashowasmallvibrationaldependence.However,forthe 1250–3000Kdata,thereislittlevibrationaldependence.Theplot forthe

ν

2 sequence (Fig.7b) showsthe200 Kand296 Kdata demonstratingasmallvibrationaldependenceandthedataatand above700Kshowingalmostnovibrationaldependence.

The data from the MCRB calculations consists of half-width and line shift parameters at the 13 temperatures of the study (200,225,275,296,350, 500,700,1000, 1250,1500,2000,2500, and 3000 K) for each unique rotational transition for the rota- tion,

ν

1, 2

ν

1, 3

ν

1, 4

ν

1,

ν

2,2

ν

2,3

ν

2, 4

ν

2,

ν

3, 2

ν

3, 3

ν

3,and 4

ν

3

bands. The data at the various temperatures was processed by the Gamache-Vispoeltemperature dependencemodel [72], called adoublepowerlaw(DPL),separatelyfortransitionsofeachvibra- tional band. The resulting temperaturedependence data and the lineshapedatawereputintoanupdatedversionofdatabasesde- velopedbyGamacheandHartmann[41,42]wherethetemperature dependenceof

γ

^{and now}

δ

^{are givenby the4 variables of the}

DPL.Thesedatabasesarelaterusedtoaddlineshapeinformation tolinefiles.

4. Predictionalgorithm

TheMCRBcalculationsmadeinthisworkareforsome94,536 ro-vibrational transitions for the principal isotopologue of water, H₂¹⁶O. The mass dependence of the line shape parameters was presentedbyLamourouxetal.[73].Forthethreeprincipalisotopo- loguesof watervapor, H₂¹⁶O, H₂¹⁸O, andH₂¹⁷O, the mass effect is very smallso the calculations presentedhere can be used for all three isotopologues. The problem is that HITRAN2020 update contains329439 transitionsfor thefirst three isotopologuesbe- longing to448 vibrationalbands. Thus,line shapedata formany thousandsoftransitionsnotstudied hereare neededto complete thedatabase.GamacheandHartmann[55] developedformulasto predict the half-width andline shift based on the semi-classical formalismofRobertandBonamy[56].Themethodwasappliedby Jacquemartet al.[57], howeverthe data were noisylimiting the procedureto~20%accuracy.Themethodwasfurtherdevelopedby GamacheandLamourouxtopredictthehalf-width,itstemperature dependence,andthelineshiftforCO₂incollisionwithN₂,O₂,air,

4 B. Vispoel, J.H. Cavalcanti and E.T. Paige et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 253 (2020) 107030

Fig. 2. MARVEL term values for the J = 10 (top panel), 11 (middle panel), and 12 (bottom panel) rotational states for the (400) vibrational state versus Ka. The + symbols are for states with Ka + Kc odd and the x symbols are for states with Ka + Kc even; MARVEL data use the color blue for odd states and red for the even states, black symbols label data augmenting the MARVEL list; see text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. VTET term values for the J = 10 (top panel), 11 (middle panel), and 12 (bottom panel) rotational states for the (400) vibrational state versus Ka. The + symbols are for states with Ka + Kc odd and the x symbols are for states with Ka + Kc even; VTET data use the color blue for odd states and red for the even states. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Half-widths in cm ^-1atm ^-1units determined using the VTET term values ver- sus the number of quanta exchanged in the transitions for νκ= 0 to 4 for ν1, ν2, and ν3; the rotation band is given by a magenta solid star symbol ( νκ= 0), the ν1sequence is given by the blue open circle symbol, the ν2sequence is given by the red asterisk symbol, and the ν3sequence is given by the black solid cir- cle symbol. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

andCO₂[62].Thesedataweremuchbetterandthepredictionrou- tinepredictedtheparameterswithlessthan1% uncertaintyfrom thealgorithm.Morerecently,Gamacheetal.[63]usedthemethod todevelopapredictionalgorithmfortheH₂O-H₂ collisionsystem with predictedhalf-widths at ~5%uncertainty. In this work they hadtofitthestretchmodesandthebendmodeseparately.Inthe last twoexamples,the algorithmswere abletofilltheentireline list fromthe HITRAN2016databasefor H₂¹⁶O, H₂¹⁸O, H₂¹⁷O, and

12C¹⁶O₂.

FromFigs.5and6,itisclearthebendmodeshouldbefitsep- aratelyfromthestretchmodes.However,therearetoomanycases showing the symmetric stretch and asymmetric stretch modes shouldbefitseparatelyaswell,unlikethestudyofRef.[63].Thus, inthisworkfitsweremadetothe

ν

1,

ν

2,and

ν

3sequencessepa-

ratelyforthehalf-widthandlineshiftforarovibrationaltransition (

ν

1’

ν

2’

ν

3’)-(

ν

1”

ν

2”

ν

3”)f←ibytheformulas

γ

v

1

v

2

v

3

v

1

v

2

v

3,f←i

=

γ

f←⁰i+Af←i[ck

ν

k]^pγf←i (3)

δ

v

1

v

2

v

3

v

1

v

2

v

3,f←i

=

δ

⁰f←i+Bf←i[ck

ν

k]^pδf←i (4)

where

γ

_f⁰_←i^and

δ

⁰_f←i^{aretheinterceptsofthefits,c}karethecoef- ficientsofthevibrationaldependenceofthepolarizability,

ν

kare thenumberofquantaexchangedinthetransition(k=1,2,or3), A_f←iandB_f←iaretheslopecoefficientsfor

γ

^and

δ

^,andpγ^f←iand p_δf←iarethepowersfor

γ

^and

δ

^{.TheQuantumCoordinate(QC)is}

givenby[c_k

ν

κ]^p.Notethecoefficientsc_karefromthevibrational dependence of the polarizability; (0.29

ν

1 + 0.03

ν

2 + 0.28

ν

3), e.g. for the symmetric stretch the QC equals (0.29

ν

1)^p. Thus, forany mode,the fit equation looks like y = a + b QC; a straightline.Thefitmethodissimilartopreviousstudies[62,63]; herethepoweris varied overa rangefrom0.3to7.0insteps of 0.002for

γ

^{and 0.3to 4.0insteps of 0.0015for}

δ

^{.At each step}

thecoefficientsA_f←i andB_f←iare determined by a linearregres- siontothehalf-widthorlineshiftdataandthesolutionwiththe absolute value of the correlation coefficient, R, closest to one is chosen.Recall, when |R| equalsone, itimpliesthe modelfits the dataperfectly.There arecases,wherethe vibrationaldependence ofthehalf-widthorlineshiftisverysmall.Forsuch casestheal- gorithmisattemptingtofitastraightlineusingapowerlawform resultinginapoorfit(smallabsolutevalue ofthecorrelationco- efficient).Fitsare madeforalltransitions,butincaseswherethe absolutevalue ofthe correlation coefficient islessthan 0.75and thehalf-widthvariesbylessthan3percentorthelineshiftvaries lessthan 0.001 cm⁻¹ atm⁻¹ over the 0-4vibrational quanta ex- changefor

ν

κ,theinterceptisset totheaveragevalue andthe slopeandpoweraresettozero.Thefitsweredoneseparatelyfor each of the 13 temperatures of the study. The fits are made for each uniquerotational transition, thedata arethe MCRB calcula- tionsforthe13vibrationalbandsofthestudy,theresultsarethe intercepts,coefficients,andpowersfor

γ

^and

δ

^{allowingthe pre-}

Fig. 5. Half-widths (top panel) and line shifts (bottom panel) in cm ^-1atm ^-1units for the (40 0)-(0 0 0) 12 012← 11 29transition determined using the VTET term values versus the number of quanta exchanged in the transitions for νκ= 0 to 4 for ν1, ν2, and ν3; the rotation band is given by a magenta solid star symbol ( νκ= 0), the ν1

sequence is given by the blue open circle symbol, the ν2sequence is given by the red asterisk symbol, and the ν3sequence is given by the black solid circle symbol.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6 B. Vispoel, J.H. Cavalcanti and E.T. Paige et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 253 (2020) 107030

Fig. 6. Half-widths (top panel) and line shifts (bottom panel) in cm ^-1atm ^-1units for the (40 0)-(0 0 0) 9 63 ← 8 62transition determined using the VTET term values versus the number of quanta exchanged in the transitions for νκ= 0 to 4 for ν1, ν2, and ν3; the rotation band is given by a magenta solid star symbol ( νκ= 0), the ν1

sequence is given by the blue open circle symbol, the ν2sequence is given by the red asterisk symbol, and the ν3sequence is given by the black solid circle symbol.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dictionofthelineshapeparameters foranyvibrationaltransition andtheirtemperaturedependence.

Figs. 8-10 show the resulting fits for the half-width and line shiftat296Kforthe

ν

1,

ν

2,and

ν

3 sequence,respectively.

Thetransitionschosen are goodrepresentative cases.Fig.8 isfor the

ν

1 sequence forthe 15106 ←15 511 rotational transition;

Fig. 9 is for the

ν

2 sequence for the 7 ₃₅ ← 6 ₆₀ rotational transition; and Fig. 10 is for the

ν

3 sequence for the 20 ₅₁₆

←20 ₂₁₉ rotationaltransition. The bluesolid circle symbolsare theMCRBdata,theredlineisthefinalfitofthedata,andbelow

theslope,power, andcorrelation coefficientare reported.The fits are quite good as shownby the plotand correlation coefficients ofthe fits. Note,the correlation coefficientsforthe lineshift fits are better than those for the half-width fits, as in other studies [62,63].

The resultsofthefits (intercept,slope,andpower) were used to develop a predictionalgorithm for H₂O-N₂ line shape param- eters. Tomake the code simpler,the results ofthe fits were put into directaccess filesforeach vibrational exchange andat each temperature. The prediction algorithm readsthese dataand uses

Fig. 7. Continued

Fig. 7. Temperature dependence of the vibrational dependence of H 2O-N 2half-width (top panel) and line shift (bottom panel) for the 7 61 ← 6 42rotational transition.

The data are at 200 K (black asterisk), 296 K (red open circles), 700 K (green open squares), 1250 K (blue open triangles), 20 0 0 K (orange solid circles), and 30 0 0 K (purple stars). Fig. 7 a is for the ν1series, Fig 7 b is for the ν2series, and Fig 7 c is for the ν3series. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Eqs.(3) and(4) topredicttransitions withasinglenormalmode vibrationaltransition.Aweightedalgorithmisused,

γ

predicted

( ν

^1,

ν

^2,

ν

³

)

^=wSS

γ

pred^ν1icted + wBM

γ

pred^ν2icted

+wAS

γ

pred^ν1icted

(5)

δ

predicted

( ν

1,

ν

2,

ν

3

)

=wSS

δ

pred^ν1icted +wBM

δ

pred^ν2icted

+wAS

δ

^νpred¹icted

(6)

where the weights for the symmetric stretch, bend mode, and asymmetricstretcharedefinedby

wSS=0.29

ν

1/

(

^0.29

ν

1+0.03

ν

2+0.28

ν

3

)

w_BM=0.03

ν

2/

(

^0.29

ν

1+0.03

ν

2+0.28

ν

3

)

wAS=0.28

ν

^3/

(

^0.29

ν

^1+0.03

ν

^2+0.28

ν

³

)

⁽⁷⁾

Note: when the transition under consideration involves only oneofthenormalmodevibrations,Eqs.(5)and(6)areequivalent toEqs.(3)and(4),respectively.

The uncertainty of the predicted half-widths and line shifts was examined by comparing the MCRB calculated values with

8 B. Vispoel, J.H. Cavalcanti and E.T. Paige et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 253 (2020) 107030

Fig. 8. Fit of the prediction equations for the half-width (top panel) and line shift (bottom panel) to the MCRB data for the ν1sequence for the 15 106← 15 511rotational transition of H 2O in collision with N 2at 296 K. Plotted are the MCRB data, the blue solid circles, and the fit, the red solid line, versus the Quantum Coordinate; here (0.29 ν1) ^p. The final slopes and powers of the fit and the correlation coefficients are top panel 0.00594, 2.638, 0.99976 and bottom panel -0.03087, 1.010, -0.99925.

Fig. 9. Fit of the prediction equations for the half-width (top panel) and line shift (bottom panel) to the MCRB data for the ν2sequence for the 7 35← 6 60rotational transition of H 2O in collision with N 2at 296 K. Plotted are the MCRB data, the blue solid circles, and the fit, the red solid line, versus the Quantum Coordinate; here (0.03 ν2) ^p. The final slopes and powers of the fit and the correlation coefficients are top panel -0.02983, 0.824, -0.99870 and bottom panel -0.10161, 1.383, -0.99702. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Fit of the prediction equations for the half-width (top panel) and line shift (bottom panel) to the MCRB data for the ν3sequence for the 20 516 ← 20 219

rotational transition of H 2O in collision with N 2at 296 K. Plotted are the MCRB data, the blue solid circles, and the fit, the red solid line, versus the Quantum Coordinate;

here (0.28 ν3) ^p. The final slopes and powers of the fit and the correlation coefficients are top panel 0.01436, 3.396, 0.99977 and bottom panel -0.04852, 1.062, -0.99950.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the values predicted by Eqs. (5) and (6). Recall [64] that the MCRBcalculationsagreewiththehighqualitylineschosentode- termine the intermolecular potential with a ~2%%difference and 3% standard deviation. The comparisons were done for the 13 bands ofthisstudy.To test thealgorithm oncombination bands, additional calculations were made for a small (325 for most bands and 588 for the 3

ν

1+

ν

3) number of transition for the

ν

2+

ν

3,

ν

1+

ν

2,

ν

1+

ν

2+

ν

3,

ν

2+2

ν

3,

ν

1+

ν

2+2

ν

3,2

ν

2+2

ν

3,

ν

1+

ν

3, 2

ν

1+

ν

3, 2

ν

1+2

ν

2+

ν

3, and 3

ν

1+

ν

3 bands. The statistics of the comparisons are shown in Table 1. The Table lists the vibra- tional band, number of transitions compared, the line shape pa- rameter, the average percent difference (APD) or average differ- ence (AD), the average deviation (ADev) of the percent differ- ence or the difference (defined by ADe

v

_=i

|

^Xi−X_are

|

/n_i), and the standard deviation of the half-width and line shift. For the 13 bands of this study, the half-width APDs are around zero and the ADev generally around 1-2 percent, with the

ν

3 se- quence where they reach 6% for

ν

3=4. The average deviations for the line shifts range between 0.0001 to 0.0006 cm⁻¹ atm⁻¹ andtheADevs rangefrom~0.0006to0.003cm⁻¹ atm⁻¹.There- sultsforthecombinationbandsdemonstratethattheweightedal- gorithm workswell givingstatisticssimilarto theother13bands of thisstudy.Fig. 11showsthe comparisonof thepredicted line shape parameters withthe MCRBcalculatedvalues forthe(010)- (000) band at 296 K. The top panel shows the percent differ- enceforthehalf-width,bluesolidcirclesymbols,versusthehalf- widthin cm⁻¹ atm⁻¹,thebottom panel showstheaverage devi- ation for the lineshift, bluesolid circlesymbols, versus the line shift incm⁻¹ atm⁻¹. The plotsfor theother 22 bandswere also

madeandareavailableatthewebsiteofthecorrespondingauthor (faculty.uml.edu/Robert_Gamache).

Becausepredictiondataareavailableforthelineshapeparame- tersfortemperaturesfrom200to3000K,thetemperaturedepen- denceof

γ

^and

δ

^{is alsodetermined inthe prediction algorithm}

viatheGamache-VispoelDPLmodel[72].

Finally,Figs. 12 and 13 show how the MCRB calculationsand MCRBpredictionscomparewiththemeasurementdataforthe

ν

2

band.Plottedare thehalf-widths(Fig.12) andlineshifts(Fig.13) in cm⁻¹ atm⁻¹ versus a linecount by transition ordered on the magnitude of the MCRB half-width or line shift. The blue open circleswitherrorbars areforthe measurementvalues,the black solid circlesymbolsarethe MCRBcalculations, andthe redopen diamond symbols are the predicted values. As can be seen the MCRBandthe predictedvaluesagree verywell withthe average ofthemeasuredvalues.

5. H₂O-N₂ linefile

AlinefileofthefirstthreeprincipalisotopologuesofH₂Owas created based on the 2020 update to the HITRAN2016 database.

The line shape data come from a number of sources: the data fromthe intercomparison database[42],from the MCRB calcula- tions,the predicteddata,andfinally,when theattributionof the transition is not in HITRAN (this occurs when ab initio data are used since only Jand symmetry are good quantum numbers), J”

averagevaluesfor

γ

^{areused.Note:theaverageofthe}half-widths asa functionof J” aredone forP-, Q-, andR-transitions andthe resultingmaximumerrorsare verylarge,1038%, 921%, and849%, respectively.Priorityisgiventotheintercomparisondata,followed

10 B. Vispoel, J.H. Cavalcanti and E.T. Paige et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 253 (2020) 107030 Table 1

Comparison of predicted line shape parameters with the MCRB calculated values.

Band # transitions γ δ Percent Error Average deviation cm ⁻¹atm ⁻¹

| Percent Error |

|Average deviation|

cm ⁻¹atm ⁻¹

Standard deviation

(000)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.10 0.000006

1.20 0.000256

2.47 0.000595 (100)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.36

0.000320

1.43 0.000923

2.90 0.001562 (200)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.07

-0.000329 1.73

0.001129 3.57

0.002109 (300)-(000) 7272 γ^∼= ∼= ∼= ∼= δ 0.40

-0.000589

2.18 0.001864

4.15 0.003456 (400)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.31

0.000403

1.04 0.001122

2.12 0.002668 (010)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.23

0.000068

1.22 0.000404

4.60 0.000613 (020)-(000) 7272 γ^∼= ∼= ∼= ∼= δ ^-0.46

-0.000136

1.73 0.000604

5.26 0.001133 (030)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.25

-0.000198

1.85 0.000846

5.32 0.001527 (040)-(000) 7272 γ^∼= ∼= ∼= ∼= δ 0.67

0.000203

1.46 0.000562

4.71 0.000992 (001)-(000) 7272 γ^∼= ∼= ∼= ∼= δ ^-0.87

0.000372

1.90 0.001438

3.11 0.002512 (002)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -0.14

-0.000370

2.65 0.002010

4.67 0.003864 (003)-(000) 7272 γ^∼= ∼= ∼= ∼= δ 0.77

-0.000588

3.77 0.002773

6.37 0.004790 (004)-(000) 7272 γ^∼= ∼= ∼= ∼= δ -1.87

-0.000447 6.32

0.003278 13.35

0.005700 (011)-(000) 325 γ^∼= ∼= ∼= ∼= δ -1.62

-0.001003

1.25 0.000807

1.87 0.001076 (110)-(000) 325 γ^∼= ∼= ∼= ∼= δ -2.33

0.004466

1.47 0.00164

2.05 0.001988 (111)-(000) 325 γ^∼= ∼= ∼= ∼= δ -0.67

-0.003825

0.85 0.001316

1.43 0.002064 (012)-(000) 325 γ^∼= ∼= ∼= ∼= δ ^-1.02

-0.000766

0.63 0.000577

0.94 0.000873 (112)-(000) 325 γ^∼= ∼= ∼= ∼= δ -0.20

-0.004262

0.65 0.000895

0.95 0.001455 (022)-(000) 325 γ^∼= ∼= ∼= ∼= δ -1.81

-0.001480

1.08 0.001111

1.57 0.001642 (101)-(000) 325 γ^∼= ∼= ∼= ∼= δ 0.77

-0.003721 1.03

0.001185 1.43

0.001632 (201)-(000) 325 γ^∼= ∼= ∼= ∼= δ 0.87

-0.004296

0.89 0.001082

1.48 0.001643 (221)-(000) 325 γ^∼= ∼= ∼= ∼= δ -0.91

-0.004618

0.91 0.001103

1.24 0.001765 (301)-(000) 588 γ^∼= ∼= ∼= ∼= δ 1.57

-0.004332 1.43

0.001211 3.00

0.002119

by single measurement values. When the measurement data are notavailable,thenextpriorityistheMCRBcalculations, thenpre- dictionvaluesandJ” averages.Thetemperaturedependenceofthe half-widthandlineshiftusetheMCRBorpredictedvaluesforall transitionswith quantum numbers; forlines without attribution, the temperature dependence of the half-width is the J” average singlepowerlawvalueandnovalueisreportedforthelineshift.

Pythonroutineswere writtento takeeach set ofdataandcreate dictionarieswiththevibrationalandrotationalquantumnumbers askeys.Usingthesedictionaries,apythoncodethenaddsthedata to the HITRAN line file of H₂¹⁶O, H₂¹⁸O, and H₂¹⁷O transitions.

Table2presents thenumbers oftransitionsthat use eachsource ofdata.Thefinalfilecontains261994,39901,and27544transi- tionsforH₂¹⁶O, H₂¹⁸O,andH₂¹⁷O,respectively.Thereferenceand error codes for the half-width, its temperature dependence, and thelineshift arefor nitrogenasthebuffer gas.Note, theformat isdifferentthantheclassicHITRANformatastheDPLformulation ofthetemperaturedependenceofthehalf-widthandlineshiftis used.Supplementalfilesofthedata(H2O-N2_2020.par),therefer- enceinformation(references_H2O-N2_2020.pdf),andadescription

Table 2

Statistics of sources of the line shape parameters of the H 2O-N 2line file.

Source

Number of transitions with line shape parameters from the states source H 216O H 218O H 217O

γintercomparison 503 459 443

δintercomparison 150 147 137

γsinglemeasurement 5 679 2 943 1 872

δsinglemeasurement 5 630 2 616 1 490

γMCRBcalculations 26 763 10 113 6 033

δMCRBcalculations 27 024 10 390 6 316

γpredicted 136 351 23 073 12 300

δpredicted 131 427 21 169 11 324

γJ” average 97622 5 579 8 277

(read_me.txt)theformatareavailable atthewebsiteofthecorre- spondingauthor(faculty.uml.edu/Robert_Gamache)andinthesup- plementalinformationofthejournal.

Fig. 11. Comparison of the predicted line shape parameters with the MCRB calculated values for the (010)-(0 0 0) band at 296 K. Top panel: Percent difference between MCRB and predicted half-widths (blue solid circle symbols) versus the half-width in cm ^-1atm ^-1; Bottom panel: Average deviation between the MCRB and predicted line shifts (blue solid circle symbols) versus the line shift in cm ^-1atm ^-1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. Half-widths in cm ^-1 atm ^-1for transitions in the ν2band from the measurement database (blue open circle symbols with measurement error bars), the MCRB calculated values (black solid circle symbols), and values from the prediction algorithm (red open diamond symbols) versus a line count ordered on the magnitude of the MCRB half-widths. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6. Conclusion

MCRBcalculationsofthehalf-widthandlineshiftweremadeat 13 temperaturesbetween200and3000Kfor7272transitionsin thebandsfor

ν

k =0-4,k=1,2,3.Theresultingdataallowthevi- brationaldependenceofthelineshapeparameterstobestudiedas a functionofrotationaltransitionandtemperature. Similartothe study done on the H₂O-H₂ collision system [63], the vibrational dependencedecreaseswithincreasingtemperatureandisnegligi-

bleabove700K.Giventhedatafor0to4

ν

1 or

ν

2 or

ν

3 quanta exchanged in the rovibrational transitions, fits of the vibrational dependencemodelofGamacheandHartmann[55] weremadeto develop a prediction routine for the half-widths, line shifts, and their temperaturedependence.Thesedatawere thenused tode- velopa predictionalgorithm. The H₂O-N₂ measurement database [42],theMCRBcalculationsdone inthiswork,andtheprediction algorithmwerethenusedtocreatealinelistfortheH₂O-N₂ col- lisionsystembasedonthe2020updatetotheHITRAN2016water