Evaluation of different parameterizations of temperature dependences of the line-shape parameters based on ab initio calculations: Case study for the HITRAN database

HAL Id: hal-02366715

https://hal.archives-ouvertes.fr/hal-02366715

Submitted on 26 Feb 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License

Evaluation of different parameterizations of temperature dependences of the line-shape parameters based on ab initio calculations: Case study for the HITRAN database

N. Stolarczyk, Franck. Thibault, H. Cybulski, H. Jóźwiak, G. Kowzan, B.

Vispoel, I.E. Gordon, L.S. Rothman, R.R. Gamache, P. Wcislo

To cite this version:

N. Stolarczyk, Franck. Thibault, H. Cybulski, H. Jóźwiak, G. Kowzan, et al.. Evaluation of different

parameterizations of temperature dependences of the line-shape parameters based on ab initio calcu-

lations: Case study for the HITRAN database. Journal of Quantitative Spectroscopy and Radiative

Transfer, Elsevier, 2020, 240, pp.106676. �10.1016/j.jqsrt.2019.106676�. �hal-02366715�

ContentslistsavailableatScienceDirect

Journal of Quantitative Spectroscopy & Radiative Transfer

journalhomepage:www.elsevier.com/locate/jqsrt

Evaluation of different parameterizations of temperature dependences of the line-shape parameters based on ab initio calculations: Case study for the HITRAN database

N. Stolarczyk

^a,∗

, F. Thibault

^b

, H. Cybulski

^c

, H. Jó ´zwiak

^a

, G. Kowzan

^a

, B. Vispoel

^d

, I.E. Gordon

^e

, L.S. Rothman

^e

, R.R. Gamache

^d

, P. Wcisło

^a

aInstitute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Torun, Grudziadzka 5, Torun 87–100, Poland

bUniv Rennes, CNRS, IPR (Institut de Physique de Rennes)-UMR 6251, F-350 0 0 Rennes, France

cInstitute of Physics, Kazimierz Wielki University, Plac Weyssenhoffa 11, Bydgoszcz 85-072, Poland

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

eHarvard-Smithsonian Center for Astrophysics, Atomic and Molecular Physics Division, Cambridge, MA, USA

a rt i c l e i n f o

Article history:

Received 22 July 2019 Revised 23 September 2019 Accepted 25 September 2019 Available online 26 September 2019 Keywords:

Temperature dependence of spectroscopic line-shape parameters

Pressure broadening and shift HITRAN Database

Molecular collisions Spectral line shapes

a b s t r a c t

Temperaturedependencesofmolecularline-shapeparametersareimportantforthespectroscopicstudies oftheatmospheresoftheEarthandotherplanets.Anumberofanalyticalfunctionshavebeenproposed ascandidatesthatmayapproximatetheactualtemperaturedependencesoftheline-shapeparameters.

Inthisarticle,weuseour abinitiocollisional line-shapecalculationsfor severalmolecularsystemsto comparethefourtemperatureranges(4TR)representation, adoptedintheHITRANdatabase[J.Quant.

Spectrosc.Radiat.Transfer2017;203:3]in2016,withthedouble-power-law(DPL)representation.Besides thecollisionalbroadeningandshiftparameters,weconsideralsothemostimportantline-shapeparame- tersbeyondVoigt,i.e.,thespeeddependenceofbroadeningandshiftparameters,andrealandimaginary partsofthecomplexDicke parameter.We demonstratethat DPLgivesbetteroverallapproximationof thetemperaturedependencies than4TR. Itshould beemphasizedthat DPLrequires fewerparameters anditsstructureismuchsimplerandmoreself-consistentthanthestructureof4TR.Werecommendthe usageofDPLrepresentationinHITRAN,andpresentDPLparametrizationforVoigtandbeyond-Voigtline profilesthatwillbeadoptedintheHITRANdatabase.WealsodiscusstheproblemoftheHartmann-Tran profile parametrizationinwhich thecorrelation parameter, η^{,and frequency ofthe} velocity-changing collisionsparameter,νvc,divergestoinfinitywhencollisionalshiftcrosseszero;werecommendasimple solutionforthisproblem.

© 2019TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Knowledge of accurate molecular spectroscopic parameters is essential forinterpreting and modeling spectra of terrestrial and planetary atmospheres, including those of exoplanets or even those of brown dwarfs. These parameters are provided in the HITRAN [1] and HITEMP [2] molecular spectroscopic databases.

Broadeningandshiftofspectral linesdueto thecollisionsoftar- get moleculeswith those ofambient gasesare amongimportant parameters that needto be usedin theopacity calculations. Tra-

∗ Corresponding author.

E-mail addresses: NikodemStolarczyk319@gmail.com (N. Stolarczyk), Robert_Gamache@uml.edu (R.R. Gamache),piotr.wcislo@fizyka.umk.pl (P. Wcisło).

ditionally, the HITRAN databaseprovided the single temperature exponent to the power law, in order to extrapolate the value of the collisional half-width at half maximum from the reference value given at 296 K to any other temperature. This approach works fairly well within the temperature ranges encountered in the terrestrial atmosphere. Nevertheless, it is now well known that the power law with a singleexponent doesnot work over a broader range of temperatures (see for instance [3–8]). There- fore,ifone aims tomodel such parameters at very diversetem- peraturesthatareencounteredindifferentplanetaryatmospheres (for instance an average temperature on Uranus is about 55 K whereas those encountered on hot Jupiters [9] or lava planets [10]canreach1000-2500K),moresophisticatedmodelshaveto be employed. Combustion applications also requiremore flexible solutions[5,7].

https://doi.org/10.1016/j.jqsrt.2019.106676

0022-4073/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

2 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Anumberofapproximationshavebeenproposedascandidates that may represent the actual temperature dependences of the broadening,

γ

0, and shift,

δ

0, parameters, see the thorough dis- cussionintheIntroduction inRef.[4].One wayofimproving the singlepower-lawapproachforpressurebroadeningwastosplitit intofourtemperatureranges(4TRrepresentation)[3];inthecase of the shift parameter, a linear function was split into the four ranges[3].The4TR approachwasadoptedinthe 2016edition of theHITRANdatabase[1].

Following previous works [11–14], a double-power-law (DPL) was recently proposed to approximate the temperature depen- dencesofboth thebroadeningandshift[4],seealsoRef [15]for thereviewofthemodernline-shapetheory.Itwasshownthatthe DPL reproduces the temperature dependences much better than thesinglepower-law functionfor broadeningandlinearfunction forshift[4].

Inthispaper,wedirectlycomparethe4TRandDPLapproaches forseveralmolecularsystems.Forthispurpose,weuseourabini- tio quantum scattering and semiclassical line-shape calculations.

Weobservethatforthecaseof

γ

0 thetwomethodsworkequally well,whilefor

δ

0theDPLapproximationisbetter.Ithastobeem- phasized,however, that forthese two line-shape parameters (

γ

0

and

δ

0)DPLrequirestwotimesfewerparametersthan4TR.

Besides the two basic line-shape parameters (

γ

0 and

δ

0), we also performed the comparison between the 4TR and DPL ap- proachesforthe mostimportantbeyond-Voigt line-shapeparam- etersdescribingthespeeddependenceofpressurebroadeningand shift (

γ

2 and

δ

2) and the velocity-changing collisions (

ν

opt^r , and

ν

optⁱ ).Concerning

γ

2and

δ

2parameters,the4TRapproachadopted in HITRAN2016 [1,3] assumed constant values of these param- eters in each of the temperature ranges. In this case, the DPL approach gives a much better approximation despite that it re- quires the same number of parameters as 4TR. With respect to the

ν

opt^r and

ν

optⁱ parameters, the approach adopted in HI- TRAN2016 [1,3] assumed a simple single power-law function;

hence,the DPLapproach introduced hereobviouslygivesamuch betterapproximation. Inthispaper, we alsoargue that,fromthe perspectiveoftemperaturedependencerepresentation,itisbetter touse the directapproach (

ν

opt^r and

ν

optⁱ ) instead ofthe

ν

vc and

η

parametrizationintroducedintheHartmann-Tranprofile[16](

η

and

ν

^{vcdivergetoinfinitywhenthe}collisionalshiftcrosseszero).

We show thatDPL givesa betteroverall approximationofthe temperaturedependencies of the six line-shape parameters than 4TR,emphasizingthatDPLrequiresfewerparametersanditsstruc- tureismuchsimplerandmoreself-consistentthanthestructureof 4TR.Werecommend the usage ofDPL representationinHITRAN, andpresentDPL parametrizationforVoigt andbeyond-Voigt line profilesthatwillbeadoptedintheHITRANdatabase.

In Sections2 and 3, we discuss the methodology and details ofthefullyquantumandsemiclassicalcalculations,respectively.In Sections4–6,weshow thecomparisonofthetwoapproachesfor the (

γ

0 and

δ

0), (

γ

2 and

δ

2) and (

ν

opt^r and

ν

optⁱ ) parameters. In Section7,we demonstratethedefinitionofthenamesoftheDPL parameterthatwillbeadoptedinHITRAN.

2. Abinitioquantumline-shapecalculations

In this section, we briefly describe the calculations based on theclose-couplingschemeandwegivesomedetailsofthepoten- tialenergysurfaces(PESs)forsystemsconsideredhere.Inthecase ofdiatomic molecules,an interaction energy,in general, depends ontheangularorientationofthecollisionalpartners, distancebe- tween their centers of mass (the intermolecular distances), and, ifapplicable,intramoleculardistances. Forthepurpose ofsolving theclose-coupling equations, the dependenceon the intermolec- ular distance is extracted from a multidimensional PES. This is

done intwo steps.First,thePESisexpandedinthecompleteba- sis of orthonormal angular functions, either bisphericalharmon- ics (see, for example Eq. (4) in Ref. [17] and Appendix therein) or Legendre polynomials (see Eq. (1) in Ref. [18]), if two collid- ing diatomic molecules ora diatomic molecule collidingwithan atom are concerned, respectively. The coefficientsof such an ex- pansion depend only on the intra- andintermolecular distances.

Second, the intramolecular dependenceof the interaction energy isreducedbyprojectingtheobtainedcoefficientsonthewavefunc- tionsofthecollidingpartners(seeEq. (2)inRef.[18]andthedis- cussion therein).The numberof resultingradial terms,employed inthefurthercalculations,dependsontheinvestigatedsystem(see Sections2.1–2.4). Theseterms are usedduring solving theclose- couplingequations[19].Boundaryconditionsonthewavefunctions of a scattering system relate the solutions of the close-coupling equations with the elementsof the scatteringmatrix (S-matrix).

The S-matrix itself is used in the calculations ofthe generalized spectroscopiccrosssections,

σ

_λ^q(

v

i,j_i,

v

f,j_f,j₂;E_kin)^{[20–24].Here,} v_i,v_f,andj_i,j_fare,respectively,theinitialandfinalvibrationaland rotationalquantumnumbersoftheopticallyactive molecule.j₂ is therotationalquantumnumberoftheperturbingmolecule(being equalto0whentheperturberisanatom),

λ

^{istherankoftheve-}

locitytensor,q isthetensorialrankoftheradiation-matterinter- action,andE_kin istherelativekinetic energyofthecollidingpair.

Thespeed-averagedpressurebroadening,

γ

0,andshift,

δ

0,param- eterscan be obtainedfrompurelyab initiocalculations. Theyare connectedwiththegeneralizedspectroscopiccrosssectionsbythe followingformula:

0

(

^T

)

+i 0

(

^T

)

=

( γ

⁰

(

^T

)

+i

δ

⁰

(

^T

) )

^p

= 1

2

π

^cn

v

r

j2

p_j2

(

^T

)

×

dxxe^−x

σ

0^q

v

i,ji,

v

f,j_f,j2;E_kin=xkBT

, (1) where nis the numberdensityof perturbing molecules,c is the speed oflight in vacuum, T isthe temperature,

^vr

^{is themean}

relative speed ofthe collidingpartnersand p_j2 isthe population ofthej₂-thstateoftheperturbermoleculeatthetemperatureT.

The

γ

0 and

δ

0 parameters are the speed-averaged values of pressure broadening and shift parameters. Even though the ex- act,activemolecule, speeddependence,

γ

^(v)and

δ

^{(v),canbe de-}

termined numerically with the ab initio calculations, it is diffi- cultto representtheminthe spectroscopicdatabases.Toaddress thisproblem, thespeed dependenceis expressedinterms ofthe quadraticapproximation[25–27]:

γ ( v )

+i

δ ( v )

≈

γ

⁰⁺ⁱ

δ

⁰⁺

( γ

²⁺ⁱ

δ

²

) v

²

v

²m

−3 2

, (2)

withvmbeingthemostprobable(absolute)speed.Theparameters

γ

2 and

δ

2canbeexpressedasafunctionoftemperature[28]:

γ

2

(

^T

)

+i

δ

2

(

^T

)

=

v

m

2 d d

v

j2

pj2

(

^T

) γ ( v ₎

+i

δ ( v ₎

v=vm

. (3)

Weprovidetheexactformulasfor

γ

^(v)and

δ

^{(v)inAppendixC.}

Similarly to

γ

0 and

δ

0, theDicke parameter,

ν

opt,can also be obtainedfrompurelyab initiocalculations,based onthegeneral- izedspectroscopiccrosssections[28,29]:

ν

^opt

(

^T

)

⁼

ν

^opt

(

^T

)

^p=₂

π

^1cn

v

r

^M2

j2

p_j2

(

^T

)

×

_∞

0

dxxe^−x 2 3x

σ

1^q

v

i,ji,

v

f,j_f,j2;E_kin=xkBT

−

σ

0^q

v

i,j_i,

v

f,j_f,j₂;E_kin=xk_BT

, (4)

whereM₂=_m^m_+m^p_p,withmandmpbeingthemassesoftheactive andperturbingmolecules,respectively.

Eqs.(1),(3)and(4)aregeneralexpressionsillustratinghowto calculatethetemperaturedependenceofthespectroscopicparam- eters fromfirst principles. In Sections 4–6 we discuss how well theycanbeapproximatedwiththeDPLand4TRrepresentations.

It should be noted that the fully abinitio line-shape parame- tersarebestsuitedwiththesophisticatedline-shapemodelsthat include the full ab initio speed dependence and reliable model ofthe velocity-changingcollisions(suchasbilliard-ball[30,31]or Blackmore [32] profiles). If the ab initio line-shape parameters are directly used with simple phenomenological profiles based on hard- orsoft-collisionmodels andapproximated forms ofthe speed dependencies of the pressure broadening and shift (such as the Hartmann-Tran profile [16,33,34]), one should expect ad- ditional percent-leveldiscrepancy withexperimental line profiles (forsome atypicalsystems,suchasAr-perturbedH₂,thisdiscrep- ancyis muchlarger). Depending onthemolecular system, differ- entstrategiesmaybeappliedtoreducetheseerrors.Forinstance, forsystemsexhibitinglargeDickenarrowingacorrectingfunction (called beta-correction)to the

ν

^{optparameter canbe usedto im-}

prove theperformance ofthehard-collision model[35].Recently, Hartmann proposed a general approach to this problem [36], in which the ab initio line-shape parameters are used with one of the sophisticated line-shapemodels to generatereference shapes in a wide rangeof pressures, andthen the referencespectra are fittedwithsomesimplerphenomenologicalmodel.Inthisscheme the fitted line-shapeparameters lose their physical meaning, but theshapesofmolecularlinesbetteragreewithexperiment.Similar testsweredonebefore,forinstanceseeRef.[37](inthatwork,due toalackofabinitioparameters,a sophisticatedline-shapemodel was used with parameters determined experimentally). Recently, this approach was tested for the requantized Classical Molecu- larDynamics Simulations corrected withtheuseof experimental spectra[38].

2.1. H2-He

Here,weemploytherecentlyreported,three-dimensional(3D) PES[29],whichisanextensionofthePESreportedinRef.[39].The PESfromRef.[39] wasargued[18]tobethemostreliableamong all theconsidered PESsin termsofreproducingthe experimental data. The newerPES[29] covers a larger rangeof intramolecular distancesthanthepreviousone,whichallowsustoinvestigatethe overtones.Moreover,thisPEShasbetterlongrangeasymptoticbe- havior.

ThePESwasexpandedoverLegendrepolynomials,andthenav- eragedoverthewavefunctionsofthehydrogenmoleculeintheini- tialandfinalrovibrationalstates.Thisledto2×4radialtermsper each rovibrational transition and 144 terms for purely rotational transitions(fordetailsseeSection3.3ofRef.[29]andSection2of Ref.[40]).

Theclose-coupling equationsweresolved forabout260values ofkineticenergyinarangefrom0.1upto9000cm⁻¹.Inthecal- culations we used the log-derivative/Airy hybrid propagator [41]. The log-derivative method was used for the intermolecular dis- tancesinarangefrom1to20a₀ andtheAirypropagatorforin- termolecular distances ina range from20 to 100 a₀. Due tothe fact that thePESdoesnot coupleortho andparastates,we dealt withseparate basis sets forthesetwo cases,associatedwithodd or even valuesof rotational levels j. One asymptoticclosed level waskeptthroughoutallthecalculations.

For the present study we investigated the following lines:

O(2) 0-0,O(3) 2-0,Q(1)2-0,Q(1) 4-0,Q(3) 5-0,S(0)0-0,S(0)1- 0,S(1)1-0,andS(1)9-0.

2.2.H₂-H₂

FortheH₂-H₂ systemweusedthe6DPESreportedinRef.[42]. This PES is a combination of two previous PESs: the 6D PES of Hinde[43],whichtakesintoaccountthestretchingofbothhydro- genmolecules,andthe4D PESofPatkowskietal.[44],calculated withrigidhydrogenmonomerswiththebondlengthsettothevi- brationalground-stateaveragedvalue.ThePESfromRef.[42]was expanded over the bispherical harmonics, leading to 15 terms, whichwerelateraveragedoverthewavefunctionsofthehydrogen molecule.

The close-coupling equations were solved for about 250 val- uesof kinetic energybetween 0.1and 5000 cm⁻¹. We used the log-derivative/Airy hybrid propagator[41] withthe log-derivative methodactive in the rangefrom2.5 to 30a₀ and Airypropaga- tor between 30 and 50 a₀. The calculations were performed for therotationallevelsoftheperturbingmolecule,j₂,from0upto5.

SimilartotheH₂-Hesystem,weanalyzedthetransitionsbetween theortho andpara statesseparately. We investigatedthe electric quadrupole(q=2)Q_v(1)linesofthefundamentalbandandofthe firstovertone.

2.3.CO-He

The ¹²C¹⁶O rovibrational energy levels, used in the basis for the CO-Heand CO-Ar dynamical calculations, are those provided by the HITRAN2016 database. The energy levels of CO in HI- TRAN2016arebased onLietal.[45] withsomecorrectionsasso- ciatedwiththeimplementationoftheCoxonandHajigeorgioupo- tential[46].Thedynamicalcalculationswereperformedonthe3D CO-HePESofHeijmenetal.[47].AFortran codeprovidedbyvan derAvoird[47]containsthevibrationallyaveragedPESinthefun- damentalandfirstexcitedvibrationalstatesof¹²C¹⁶O.Theresult- ingPESshavebeenexpandedover14Legendrepolynomials(upto l=13).Theclose-couplingcalculationsareverysimilartotheones performedon these PESs, in Ref. [48]and, thus, will not be de- scribed.Inordertodeducethepressurebroadeningandshiftcoef- ficientsforthepurelyrotationalandrovibrationalR(0)0-0,R(0)1- 0,R(20)0-0andR(20)1-0lines,necessary[48]S-matrixelements were computed for about 180 and 140 relative kinetic energies, rangingfrom0.1to2000cm⁻¹,fortheR(0)0-0,R(0)1-0,R(20)0- 0andR(20)1-0lines,respectively.

2.4.CO-Ar

The PES used for the calculations of CO-Ar cross sections was the purely ab initio potential calculated by Sumiyoshi and Endo[49].TheC-Obondlengthvalues,r_CO,spannedarangefrom 1.00 ˚Ato1.35 ˚Ainincrementsof0.05 ˚A.Thisrangeofr_CO covers 99.98%ofthesquaredwavefunctionamplitudeoftheuppervibra- tionalstate(

ν

=1,j=0),whatwe considersufficientforcalcula- tionsof

ν

=0→1transitions.ThePESwasvibrationallyaveraged overthe lower anduppervibrationalstates andexpandedin the basisofLegendrepolynomialswithtermsuptothen=10order.

TheS-matriceswerecalculatedwiththelog-derivative/Airyhy- bridpropagator[41]. Thelog-derivative methodwasused from2

˚Ato10 ˚AandtheAirypropagatorwasusedfrom10 ˚Atotheend ofpropagation.Theendofpropagationwassetnocloserthanthe CO-Ardistanceof 20 ˚A.It wasincreasedtotheseparation atthe furthestclassicalturningpointfoundinthecentrifugalpotentialif thepointwasfoundtobelyingfurtherthan20 ˚A.Thegridofcol- lisionenergiesatwhichtheS-matriceswereobtainedconsistedof 376 points. It waslimited by the maximal energy of1700 cm⁻¹ and the grid density was adjusted to obtain smooth

σ

₀^{q curves,}

withdenser samplingat low collision energies andsparser sam- plingathighcollisionenergies.Thelinesofthefundamentalband

4 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Fig. 1. Ab initio temperature dependences of the pressure broadening parameter γ0, (black lines) and its 4TR and DPL approximations (red and green lines, respectively).

To resolve overlapped curves below every graph we show the absolute residuals (the same color notation). The first temperature range is not covered by the semiclassical MCRB calculations. The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

forwhichthespectroscopiccrosssectionswereobtainedare:P(1), P(2),P(8),R(0),R(1),andR(7).

3. Abinitiosemiclassicalline-shapecalculations

The calculation of the half-width and line shift for the H2O- H₂, H₂O-N₂, HDO-CO₂ and CO₂-CO₂ systems were made using thecompleximplementationofthesemi-classical Robert-Bonamy formalism [50,51] with the modification suggested by Ma [52], labeled as MCRB. In this complex-valued formalism, the half- width,

γ

^{, andlineshift,}

δ

^{, fora} rovibrational transitionf←iare

givenby

0

(

^T

)

⁺ⁱ 0

(

^T

)

⁼

( γ

0

(

^T

)

⁺ⁱ

δ

0

(

^T

) )

^p

= 1 2

π

^cn

_∞

0

d

v

r

v

rf

( v

r

)

×

_∞

0

db2

π

^{b 1−e−iS1+Im(S2)J2e−Re(S2)J2}

, (5)

where vr is the relative velocity, f(vr) is the Maxwell-Boltzmann distributionofspeed,

J₂ isanaverageoverthestatesoftheper- turber,andbisthe impactparameter.S₁ andS₂ arethefirst and second order terms in the successive expansion of the Liouville

Fig. 2. Ab initio temperature dependences of the pressure shift parameter δ0, (black lines) and its 4TR and DPL approximations (red and green lines, respectively). To resolve overlapped curves below every graph we show the absolute residuals (the same color notation). The first temperature range is not covered by the semiclassical MCRB calculations. The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

scatteringmatrix, whichdependson the rovibrationalstates that areinvolved,theassociatedcollisionally-inducedjumpsfromthese states, on the intermolecularpotential, andthe characteristics of the collision dynamics. Note, semiclassicalrefers to the fact that the internal structure of the collidingmolecules is treatedquan- tum mechanicallyandthat the dynamics ofthe collision process are treatedby classical mechanics. Thus, in an optical transition from f←i, the active molecule will undergo collisions with the bathmolecules,forwhichthetrajectoriesaredeterminedbyclas- sical mechanics. Inthese collisions, the initial andfinal statesof

theradiatingmolecule, iandf,willundergo collisionally-induced transitionstostatesi’andf’,interruptingtheradiationandcausing collisionalbroadening. The statesi’ andf’are calledcollisionally- connectedstates and are givenby selection rules determined by the wavefunctions and intermolecular potential coupling terms.

Giventhe large numberofterms inthe intermolecularpotential, therearemanystates,i’orf’,thatarepossible.Thequantumme- chanicalcomponentsofthecalculationaretheenergyofthestates involvedinthecollisionally-inducedtransitionsandthewavefunc- tionsforthestates,whichareusedtodeterminetheprobabilityof

6 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Fig. 3. The relative MAD for the 4TR and DPL representations of the temperature dependence of the pressure broadening parameter γ0(see text for more details). The 1%

level is indicated with the horizontal black dashed lines. The first temperature range is not covered by the semiclassical MCRB calculations.

acollisionally-inducedtransition.Thedetailsaboutourcalculations fortheH2O-H2,H2O-N2,HDO-CO2andCO2-CO2systemsconsiderd herecanbefoundinRefs.[53–56].

4. Temperaturedependencesofthepressurebroadeningand shiftparameters

γ

0 and

δ

0

In this section, we directly compare the two representations (4TR and DPL) of the temperature dependences of the pressure broadening,

γ

0,andpressure shift,

δ

0, parameters.In Section 4.1, wediscussthedetailsofthe4TR[1,3]andDPL[4]approaches.The 4TRapproachrequireseightparameters(twoparameterspereach temperaturerange)todescribethefulltemperaturedependenceof asingleline-shapeparameter,whiletheDPLrequiresonlyfourpa- rameters(in theDPLcaseasinglefunctioncoversallthetemper- atureranges). InSection 4.2,wepresenttheresultsandthecom- parisonofthetworepresentations.

4.1.4TRandDPLrepresentationsof

γ

0and

δ

0

The 4TR representation (in the form adopted in HI- TRAN2016 [1,3]) splits the temperature dependence into four ranges,eachrepresentedbyadifferentreferencetemperature,T_ref:

1^st range:T < 100K,T_ref=50K,

2^ndrange:100K ≤T< 200K,T_ref=150K, 3^rd range:200K ≤T< 400K,T_ref=296K, 4^thrange:400K ≤ T,T_ref=700K.

Ineachofthefourranges,thetemperaturedependencesof

γ

0

and

δ

0aredescribedbythepower-lawandlinearfunctions:

γ

0

(

^T

)

=

γ

0

(

^Tref

)

× Tref

T

ⁿ

, (6)

δ

⁰

(

^T

)

=

δ

⁰

(

^Tref

)

+

δ

0

(

^T−T_ref

)

. (7) Thisrepresentationrequiresapairoffittedparameters,

γ

0(T_ref) andn,or

δ

0(^Tref) ^and

δ

0,foreachofthefourranges,resultingin eightparameters intotalforeach ofthetwo(

γ

0 and

δ

0) spectral line-shapeparameters. Additionally,we enforced the functionsto be continuousbetweenthe ranges,which resultedinthree addi- tionalconstraints.Tofittheserepresentationstotheabinitiodata, weperformedatwo-stepfittingprocedureusingtheleast-squares method.Inthefirststep,weusedthefittingalgorithmforthe2^nd and3^rd temperatureranges(mostimportantforthestudiesofat- mospheres oftheEarthandother planets),demanding theconti- nuityatT=200K.Inthenext step,we performedthefittingfor theremaining 1^st and4^th temperatureranges,demandingthefit- tedfunctions(Eqs.(6)and(7))tobecontinuousonthetwoother borders,i.e.atT=100KandT=400K.

Recently, Gamache andVispoel proposed the DPL representa- tion for

γ

0 and

δ

0 [4] (here the names of the parameters are changedtobeconsistentwithnamingforotherline-shapeparam- eters):

γ

⁰

(

^T

)

=g0

(

^Tref/T

)

ⁿ+g₀

(

^Tref/T

)

ⁿ, (8)

δ

0

(

^T

)

^=d0

(

^Tref/T

)

^m+d0

(

^Tref/T

)

^{m. (9)}

Fig. 4. The MAD for the 4TR and DPL representations of the temperature dependence of the pressure shift parameter, δ0(see text for more details). The first temperature range is not covered by the semiclassical MCRB calculations.

Fig. 5. Temperature dependence of the speed dependence of the broadening parameter, γ2. The black lines are the functions determined from our ab initio calculations, while red and green ones are the 4TR and DPL representations, respectively. To resolve overlapped curves below every graph we show the absolute residuals (the same color notation). The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

8 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Fig. 6. Temperature dependence of the speed dependence of the shift parameter, δ2. The black lines are the functions determined from our ab initio calculations, while red and green ones are the 4TR and DPL representations, respectively. To resolve overlapped curves below every graph we show the absolute residuals (the same color notation).

The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

Fig. 7. The MAD for the 4TR and DPL representations of the temperature dependence of the speed dependence of the broadening parameter, γ2(see text for more details).

Fig. 8. The MAD for the 4TR and DPL representations of the temperature dependence of the speed dependence of the shift parameter, δ2(see text for more details).

AsingleDPLfunctioncoversallthetemperatureranges.Therefer- encetemperature, T_ref,is296K.Thisrepresentationrequiresfour parameters (twotimesfewerthan4TR) torepresentthefull tem- peraturedependenceofthespectralline-shapeparameterandhas no additionalconstraints.We usedtheleastsquares algorithmto fitthisrepresentationtotheabinitiodata.

4.2. Resultsandcomparison

The tworepresentations (4TR andDPL)weretested ontheab initio results for a sample of 36 molecular lines from eight col- lisional systems. We employed the fully quantum ab initio cal- culations forthe CO-Ar, CO-He, H₂-He andH₂-H₂ systems,while the calculationsfortheH₂O-H₂,H₂O-N₂,HDO-CO₂ systems were basedontheabinitiosemiclassicalapproach.Thefulltemperature dependencesoftheeightchosenrovibrationaltransitions(onefor eachmolecularsystem)areshowninFigs.1and2forthecasesof

γ

0and

δ

0,respectively.Figs.3and4presentthecomparisonofthe maximalabsolutedifference(MAD)forbothapproximationsforall theconsideredlines.

ItisseenfromFigs.1and3that(despitethatDPLrequirestwo timesfewerparameters)bothapproximationsgivesimilaraccuracy of

γ

0 representation,andin theprioritized 2^nd and3^rd tempera- turerangesMADisatthelevelof1%inmostofthecases.

Figs. 2and4show thecomparisonof4TRandDPLforthe

δ

0

parameter.Forthetypical(monotonic)behaviorofthetemperature dependencesof

δ

0,theDPLworksbetter(CO-He,H₂O-N₂,CO₂-CO₂ andH2-H2 casesinFig.2)orsimilarly(CO-ArandHDO-CO2cases inFig.2)to4TR;itshouldbenotedthatalsoherethe4TRrequires twiceasmanyparametersasDPL.If,however,thetemperaturede-

pendencehasaminimumormaximum(e.g.,H₂-HecaseinFig.2), the4TRrevealsmoreflexibilityduetothelargernumberofthefit- tedparameters and,hence,givesabetteraccuracyoftemperature dependencerepresentation.

5. Temperaturedependencesofthepressurebroadeningand shiftspeeddependenceparameters,

γ

2and

δ

2

In this section, we shall make a similar comparison as pre- sentedinSection4.Here,wediscussthetemperaturedependence ofthespeeddependenceofthepressurebroadeningandshiftco- efficients,

γ

2 and

δ

2. We limit the discussion to the20 molecu- larlines forwhichwe performed thefully quantumcalculations.

InSection5.1,we discussthedetails of4TRapproachintheform adoptedinHITRAN2016[1,3]andDPLrepresentation[4].Boththe approximationsrequirefour parameters todescribe the full tem- peraturedependenceofthe

γ

2and

δ

2parameters.Section5.2con- tainsthecomparisonofthetworepresentationsanddiscussionof theresults.

5.1. 4TRandDPLrepresentationsof

γ

2and

δ

2

For the case of the

γ

2 and

δ

2 parameters, the form of the 4TR approximation adoptedin HITRAN2016 requires fourparam- eters[1,3];itwasassumedthat

γ

2 and

δ

2 areconstantovereach ofthetemperatureranges:

γ

2

(

^T

)

=

γ

2

(

^Tref

)

, (10)

δ

2

(

^T

)

⁼

δ

2

(

^Tref

)

^{, (11)}

10 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Fig. 9. Ab initio temperature dependences of the real part of the Dicke parameter, ν^ropt, (black lines) and its 4TR and DPL approximations (red and green lines, respectively).

To resolve overlapped curves below every graph we show the absolute residuals (the same color notation). The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

Fig. 10. Ab initio temperature dependences of the imaginary part of the Dicke parameter, νoptⁱ , (black lines) and its 4TR and DPL approximations (red and green lines, respectively). To resolve overlapped curves below every graph we show the absolute residuals (the same color notation). The residuals are in the same units as the main plots, i.e., 10 ⁻³cm ⁻¹atm ⁻¹.

whereT_reftakesoneofthefourvaluesspecifiedinSection4.1.This representationrequiresonefittedparameterpereachtemperature range,resultinginfourparametersintotal.

TheDPLapproximationsforthecaseofthe

γ

2 and

δ

2 parame- tershaveexactlythesameformasfor

γ

0 and

δ

0:

γ

2

(

^T

)

=g2

(

^Tref/T

)

^j+g₂

(

^Tref/T

)

^j, (12)

δ

²

(

^T

)

^=d2

(

^Tref/T

)

^k+d2

(

^Tref/T

)

^{k. (13)}

withthesameT_ref=296K.

5.2. Resultsandcomparison

In the cases ofthe

γ

2 and

δ

2 parameters, we performed the same comparisonsof the 4TRand DPL approachesasfor

γ

0 and

δ

0 (seeSection 4.2).Thefull temperaturedependencesof

γ

2 and

δ

2 for the same sample of the four molecular transitions as in Figs.1and2arepresentedinFigs.5and6.Figs.7and8showthe MADforallthe20analyzedtransitions.TheDPLmethodisbetter by far inalmost all the consideredcases, despitethat it requires thesamenumberoffittedparameters.

Fig. 11. The MAD for the 4TR and DPL representations of the temperature dependence of the real part of the Dicke parameter, νopt^r (see text for more details).

6. TemperaturedependencesoftheDickeparameter,

ν

^opt

The fourline-shapeparameters discussed inSections4and5, in fact, can be seen asreal andimaginary parts oftwo complex parameters.Thefirstone,

γ

⁰+i

δ

⁰,isacomplexrateofrelaxation ofopticalexcitation.Thesecondone,

γ

2+i

δ

2,quantifiesthespeed dependenceofthisrate. Similarly,therateoftheopticalvelocity- changingcollisions,

ν

^opt,(alsocalledtheDickeparameter)isalsoa complexnumberthat canbe writteninthesamemanner,

ν

opt =

ν

opt^r +i

ν

optⁱ ,where

ν

opt^r and

ν

optⁱ arecalledrealandimaginaryparts oftheDickeparameter,respectively.

It should be noted that in the Hartmann-Tran (HT) pro- file [16,33,34]the two partsof

ν

^{opt are expressedindirectly with}

theuseof

ν

^vcand

η

parameters:

ν

opt^r =

ν

vc−

ηγ

0, (14)

ν

optⁱ =−

ηδ

0. (15)

Tocalculate

η

^and

ν

vcweconvertourabinitio

ν

optparameter(ob- tainedfromEq. (4))usingthefollowingformulas:

ν

^vc=

ν

opt^r −

ν

optⁱ

γ

⁰

δ

^{0, (16)}

η

⁼⁻

ν

optⁱ

δ

^{0 . (17)}

It shouldbe notedthatthe aboverelations, Eqs.(14)–(17),ignore the speed dependence ofthe

ν

^{opt parameter which wasadopted}

inthe HTprofile.We show inAppendixBthat the HTprofileen- forcesanunphysicalformofthespeeddependenceofthecomplex Dickeparameter.WithintheHTprofileparametrizationthereisno flexibilitytoadjustthespeeddependenceof

ν

^{opttoamorephysi-}

calone.Therefore,thebestthatcanbedonetolinktheHTprofile parameters with the ab initioones is to enforce that the speed- averagedvaluesofthecomplexDickeparameterarethesame;this iswhatwedidinEqs.(14)–(17).

The same parametrization as in the HT profile was adopted in HITRAN2016 [1,3] to represent the non-Voigt line profiles. In thispaper, we argue that the more straightforward notation, i.e.

ν

^opt =

ν

opt^r +i

ν

optⁱ ,allowsonetoavoidsomeadditionaldifficulties withthe temperature dependences that may arise when the

ν

vc

and

η

parametrizationisused,seeAppendixA.Werecommendthe useoftheexplicitnotationoftherealandimaginarypartsofthe Dickeparameter, i.e.

ν

opt =

ν

opt^r +i

ν

ⁱopt, and to neglect its speed dependence.FordetailsoftheDPL representationfortheHITRAN databaseseeSection7.

6.1. 4TRandDPLrepresentationsof

ν

opt^r and

ν

optⁱ

Since the notation of the complex Dicke parameter recom- mended in this paper differs from the one adopted in HI- TRAN2016 [1,3] (i.e., we use

ν

opt^r and

ν

ⁱopt instead of

ν

^{vc and}

η

^),

herewedonotcomparetheDPLandHITRAN2016temperaturede- pendencesinadirectway(seeAppendixAforadirectcomparison).

Nevertheless,toshowtheflexibility oftheDPL approximationfor thecaseoftheDickeparameter,wecompareitwiththe4TRrep-

12 N. Stolarczyk, F. Thibault and H. Cybulski et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 240 (2020) 106676

Fig. 12. The MAD for the 4TR and DPL representations of the temperature dependence of the imaginary part of the Dicke parameter, νoptⁱ (see text for more details).

resentationthat hasthesameformasfor

γ

0 and

δ

0 discussed in Section 4 (i.e.for each temperature range,single powerlaw and linearfunctionsaretakenforrealandimaginarypartsoftheDicke parameter,respectively):

ν

opt^r

(

^T

)

=

ν

opt^r

(

^Tref

)

× Tref

T

^m

, (18)

ν

optⁱ

(

^T

)

=

ν

optⁱ

(

^Tref

)

+

ν

optⁱ

(

^T−Tref

)

. (19) Herewetakethesamereferencetemperature, T_ref,asforthe4TR for

γ

0and

δ

0,seeSection4.Moreover,followingSection4,ween- forcedthecontinuityofthefunctionontheendpointsofthetem- perature ranges and performedthe same two-step fitting proce- dure.

The DPL approachhas exactlythe sameformasfor theother line-shapeparametersdiscussedinSections4and5:

ν

opt^r

(

^T

)

=r

(

^Tref/T

)

^p+r

(

^Tref/T

)

^p, (20)

ν

optⁱ

(

^T

)

⁼ⁱ

(

^Tref/T

)

^q+i

(

^Tref/T

)

^{q, (21)}

Also, similar to theother line-shapeparameters, herea single DPLfunctioncoversallthetemperaturerangesandT_ref=296K.

6.2.Resultsandcomparison

InFigs.9and10,weshowthefulltemperaturedependencesof the

ν

^roptand

ν

optⁱ parameters determinedfromourquantum scat-

teringcalculationsandacomparisonoftheir4TRandDPLapprox- imationsforthesamefourlinesasinFigs.5and6.InFigs.11and 12,wecomparetheMADfor4TRandDPLforallthecasesconsid- eredhere.

Thetemperaturedependence oftherealpartofthe Dickepa- rameter,

ν

opt^r , usually decaysmonotonically withthetemperature and can easily be described by a single power-law function, see Fig. 9. For molecular hydrogen, it directly follows from the fact that, inthiscase, the

ν

^roptparameter is closetothe frequencyof the velocity-changing collisions (that is calculated from the dif- fusion coefficient).Inmostof thecases considered here,the 4TR reproducestheabinitotemperaturedependencesmoreaccurately thantheDPL,especiallyintheprioritized,2^ndand3^rdtemperature rangeswhere4TRreachessub-percent accuracyforalmost allthe considered transitions, see Fig. 11. Despite its higher accuracy, it needstobenotedthat4TRrequires8parameterstorepresentthe fulltemperaturedependenceofthe

ν

^roptparameter,whiletheDPL representationrequiresonly4parameters.

Thebehavior ofthetemperaturedependenceoftheimaginary partofthe Dickeparameter,

ν

optⁱ ,dependson themolecular sys- tem, see Figs.10 and12. Inthe caseswhen the temperaturede- pendenceismonotonic,both methodsprovideagoodapproxima- tionbut,usually,DPLisslightlymoreaccurate,seethetwobottom panelsinFig.10.However,ifthefunctionexhibitsatleastoneex- tremum, see the two upperpanels in Fig. 10, the 4TR approach showsmoreflexibility than DPL duetothe largernumber ofthe fittedparameters.InFigure12,wecomparetheperformanceofthe DPLand4TRfor

ν

optⁱ foralltheconsideredlines.

Table 1

HITRAN parametrization of the DPL temperature dependence of the beyond-Voigt line-shape parameters. In case of the Voigt profile only the first two parameters, γ0(T) and δ0(T), should be considered.

Description of the parameters The symbols of the line-shape parameter [cm ⁻¹/atm]

DPL parametrization of the temperature dependence of the line-shape parameters

The formulas illustrating how the parameters should be translated into the line-shape parameters ( T ref= 296 K).

Coefficient 1 Coefficient 2 Exponent 1 Exponent 2

Pressure broadening γ0(T) g 0 g ₀ n n γ0(^T)=g 0(T ref/ T)ⁿ+ g ₀(T ref/ T)ⁿ Pressure shift δ0(T) d 0 d ₀ m m δ0(T)=d 0(T ref/ T)^m+ d ₀(T ref/ T)^m Speed dependence of the

pressure broadening γ2(T) g 2 g ₂ j j γ2(T)=g 2(T ref/ T)^j+ g ₂(T ref/ T)^j Speed dependence of the

pressure shift δ2(T) d 2 d ₂ k k δ2(T)=d 2(T ref/ T)^k+ d ₂(T ref/ T)^k Real part of the Dicke parameter ν^ropt(^T) r r p p ν^ropt(^T)= r (T ref/ T)^p+ r (T ref/ T)^p Imaginary part of the Dicke

parameter νⁱopt(T) i i q q νⁱopt(T)= i (T ref/ T)^q+ i (T ref/ T)^q

7. TheDPLrepresentationfortheHITRANdatabase

The full setofthe coefficientsthatis neededtorepresentthe DPLtemperaturedependenciesofallthesixline-shapeparameters (i.e.:

γ

0,

δ

0,

γ

2,

δ

2,

ν

opt^r and

ν

^ropt)describedinSections4-6con- sistsof24coefficientsperonemolecular transition(4 coefficients per eachline-shape parameter),seeTable 1.Thestructure ofthis parametrization is simpler and requires three parameters fewer thanthe4TRparametrizationadoptedinHITRAN2016[1,3](cf.Tab.

7inRef.[3]).

8. Conclusion

In this article, we analyzed the problem of seeking an opti- mal simpleanalytical representationof temperaturedependences ofcollisionalline-shapeparametersthatcanbeadoptedintheHI- TRANdatabase.Weusedourabinitiofullyquantumandsemiclas- sical collisional line-shape calculations to compare the 4TR rep- resentation, adopted in the HITRAN database in 2016 [1], with the DPL representation.We considered two basiccollisional line- shape parameters (

γ

0 and

δ

0) that enter into the Voigt profile and fourbeyond-Voigt line-shapeparameters (

γ

2,

δ

2,

ν

opt^r ,

ν

ⁱopt).

We demonstrated that DPL gives better overall approximation of the temperaturedependences than4TR. Itshould be emphasized thatDPLrequiresfewerparametersanditsstructureismuchsim- pler andmoreself-consistent than the structureof 4TR.The DPL datasets can be easily implemented into the HITRAN relational database structure due to its flexibility [57]. We present a DPL parametrizationforVoigt andbeyond-Voigt lineprofilesthat will beadoptedintheHITRANdatabase.

We also discussed the problem of the Hartmann-Tran profile parametrization in which the correlation parameter,

η

^{, and fre-}

quency ofthevelocity-changing collisionsparameter,

ν

^{vc, diverge}

toinfinitywhenthecollisionalshiftcrosseszero.Wedemonstrate that this issuecan be resolved by using a direct parametrization with a complex Dickeparameter,

ν

opt, instead of

η

^and

ν

vc that wereadoptedintheHTprofile.

Oneshould beawarethatwhen modelingspectraofhotenvi- ronments, the usage of theHITEMP [2] databaserather than HI- TRANisstronglyadvised.TheHITRANdatabasetargetsarangeof temperaturesthatdoesnotsubstantiallyexceedthoseencountered on theEarthand withan exception ofa few diatomicmolecules (NO,OH,HF,HCl,HBr,HIandH₂)doesnotprovideenoughtransi- tions.Itisworthmentioningthatifthetemperaturerangecovered in theexperiments isnot extensiveit isbetter to fixthe param- eters associated with the second exponent termto be 0 andef- fectively usea singlepower law.Wealso recommendthat inthe

case when there is a sufficient extent of data to fit to the DPL, oneshould report thenew datainboth newDPL andtraditional single-power law formats if possible. This will allow researchers thatworkwiththeroomtemperatureenvironmenttocontinueus- ingexistingradiativetransfercodes.

Acknowledgments

We thank Jean-Michel Hartmann and Ha Tran for the dis- cussion and their remarks regarding the parameterization of the Hartmann-Tran profile and the strategy for improving accuracy when theparameters ofthe Hartmann-Tran profile are based on ab initio calculations. NS, HJ and PW contribution is supported by the National Science Centre in Poland through projects nos.

2015/19/D/ST2/02195 and2018/31/B/ST2/00720. HC acknowledges the support by the National Science Centre in Poland through project no. 2014/15/B/ST4/04551. GK acknowledges the support by the National Science Centre, Poland through project nos.

2016/21/N/ST2/00334 and 2017/24/T/ST2/00242. RRG and BV are pleased to acknowledge supportof this research by the National ScienceFoundationthroughgrantno.AGS-1622676.Anyopinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the viewsof the National Science Foundation. The projectis co- financedbythePolishNationalAgencyforAcademicExchangeun- derthePHCPoloniumprogram(dec. PPN/X/PS/318/2018).TheHI- TRANdatabaseissupportedbyNASAAURANNX17AI78GandNASA PDARTNNX16AG51Ggrants.

AppendixA. TemperaturedependencesoftheHTprofile

ν

vc

and

η

^parameters

Inthissection,wecomparetheparametrizationofthevelocity- changingcollisionsemployedinthispaperandthoseused inthe HTprofile.Followingtheconventionadoptedinthispaper,theto- talcollisionaloperator,Sˆ,thatenterstheline-shapetheorycanbe writtenas

Sˆ/p=−

( γ

⁰⁺ⁱ

δ

⁰

)

−

( γ

²⁺ⁱ

δ

²

)

^b

( v )

+

(

ν

opt^r +i

ν

optⁱ

)

^Mˆ, (A.1)

where

ν

^optisthespeed-averagedDickeparameter,seeEq.(4)(here we neglect the speed dependence of

ν

opt, see Appendix B).

b(

v

₎=(

v

²/

v

²_m−3/2),vm isthe mostprobablespeed ofan active molecule, p ispressure and Mˆ is a normalized velocity-changing operator.Dependingonthechoiceoftheline-shapeprofile,Mˆ can