biliCHsm

ulouse

a r t i c l e i n f o

Received in revised form27 August 2013Accepted 5 September 2013Available online 18 September 2013

a b s t r a c t

In some temperature and pressure conditions, for an

could be sufficient. For example in the centre part of high

ic entries and inbsorbing regionsution cannot be

s were built tow temperatures.ssure conditionsular dissociation

which increases the predominance of molecules toward

in the same way the radiative database could be restricted

Contents lists available at ScienceDirect

w.el

Journal of QuantitativRadiative T

Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 4344440022-4073/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jqsrt.2013.09.005to low vibrational levels.Themolecular database RADEN [1] covers a lot of diatomic

electronic systems but the transition probability tables arenot public. Nevertheless, some useful recommendations are

n Corresponding author. Tel.: 33 561 558 433.E-mail address: alain.gleizes@laplace.univ-tlse.fr (A. Gleizes).exhaustive description of the radiative phenomena occur-ring in the plasma, knowledge of the atomic contribution

higher temperatures. Then at the higher temperatures,some important electronic systems could be omitted andCharacterising the radiative contribution of molecularspecies in a plasma is a fundamental task in a large varietyof fields. Molecular radiation occurs in plasma devices(syngas or biofuel reactor, reaction chemistry), spectro-scopic diagnostics, astrophysics problems (atmosphericentries, stellar radiation) and meteorology compositionanalysis. This task requires an extensive and accuratedatabase of transition probabilities to describe the spectra.

ignored. However, for lower temperaturplasmas, combustion devices, atmospherthe more general case of the peripheral aof plasma devices, the molecular contribneglected.

Some molecular radiative databaseoperate at atmospheric pressure and loCritical limitations emerge for high predue to temperature shift of the molec1. Introduction intensity arcs, the molecular contribution can often bee regions in arcKeywords:COH2 plasmaDiatomic moleculesTransition probabilitiesVibrationalrotational couplingdiatomic molecules contributing to the discrete radiation of COH2 syngas plasma. Wepropose extensive tables of rovibrational transition probabilities for the main electronicsystems of OH, CH, CH , CO and CO . The rotational dependence of the nuclear wave-functions was included in our calculations to take into account the coupling between therotational and vibrational motions. References are also given to data for O2, C2 and H2molecules already published in the literature. The calculations were performed using theRydbergKleinRees (RKR) inversion procedure for the reconstruction of the potential-energy curves and an improved Numerov-type method was used to obtain the rovibra-tional wave-functions by solving the radial Schrdinger equation. We rigorously selectedthe most up-to-date equilibrium spectroscopic constants for the RKR procedure and themost accurate electronic transition moment functions (ETMF) available in the literature.The results obtained with this procedure were systematically validated by comparisonwith available experimental observations.

& 2013 Elsevier Ltd. All rights reserved.Article history:Received 31 January 2013

This paper focuses on the calculation of the radiative transition probabilities for the mainTables of radiative transition probadiatomic molecular systems of OH,occurring in COH2 syngas-type pla

T. Billoux, Y. Cressault, A. Gleizes n

Universit de Toulouse, CNRS ; UPS-INP, LAPLACE, 118 route de Narbonne, To

journal homepage: wwties for the main, CH , CO and CO

a

cedex 9, France

sevier.com/locate/jqsrt

e Spectroscopy &ransfer

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444 435in LIFBASE [3] but some electronic systems are missing formodel applications. Thus we decided to compute all thesystems of CH and CH .

We present an extensive and exhaustive molecularradiative database which takes into account all the dia-tomic contributions encountered in COH2 plasma. Toensure that it remains valid for the full range of workingconditions, our transition probability tables were extendedto the high vibrational and rotational levels to cover a largerange of temperatures (300K15kK), preventing somemolecular systems from becoming preponderant as aresult of the shift of the temperature dependant dissocia-tion rate with pressure (1100 bars). We report extensivetables of rovibrational transition probabilities for the mainmolecular systems occurring in COH2 plasma. The mainelectronic systems of the OH, CH, CH , CO and CO

molecules were investigated.We calculated the transition probabilities for the AX

violet and the XX Meinel electronic systems of OH, for theXX, AX, BX and CX electronic systems of CH, for theAX system of CH , for the XX infrared and the AXfourth positive systems of CO and for the AX Comet-Tail,the BX first negative and the BA BaldetJohnson sys-tems of CO . The electronic systems of O2, C2 and H2 werenot dealt within this paper since these electronic systemshave either already been studied in detail elsewhere or didnot appear predominant in COH2 plasma radiation. Allreferences for transition probability tables in the literatureare given in this paper.

The present paper is organised in five parts. In Section 2we briefly describe the context of the work and how thetransition probability tables can be used. In Section 3, theRydbergKleinRees (RKR) procedure for the calculation ofthe molecular potential curves is briefly described. InSection 4, we develop a Numerov-type procedure usedfor the calculation of the rovibrational wave-functions bysolving the radial Schrdinger equation. In Section 5, thetransition probabilities and the corresponding radiativelifetimes obtained with this procedure are systematicallycompared with available experimental and theoreticalresults. Finally extensive transition probability tables foreach electronic system of interest are given under electro-nic form in the online version.

2. Background

The integrated intensity due to the spontaneous emis-sion of a single molecular or atomic line is given by therelation

Aul Wm3 sr1 nu Aul hcsul

41

Using expression (1), three parameters are required tocharacterise each line transition:

The emitting population number density nu. The line position sul related to the energy difference

between the two ideal unperturbed states.mosystments to use [2]. The AX, BX and CX electronicems of CH and AX system of OH were accurately treatedgiven for the selection of the most accurate transitiontheallo The Einstein spontaneous emission coefficient Aul cor-responding to the transition between the upper and thelower rotational states involved.

The Aul expression depends on the electronic, vibra-tional, rotational, fine and hyperfine structure of themolecule [4]. We restricted this work to the calculationof the spin-allowed electronic dipolar transitions. Byinvestigating the equilibrium composition of COH2plasma covering the whole range of temperature, pressureand element ratios considered in this work, the maindiatomic molecules selected for calculations were CO,OH, CH, O2, C2, H2, CO and CH . Some high energyelectronic states are not significantly populated for thewhole range of temperature and pressure conditionsretained in this work.

We recommend the tables of Abgrall et al. [5] and ofthe Observatoire de Paris [6] for the H2 molecule as theystand since these authors accurately treated the rotationaldependence of the transition probabilities. Some higherelectronic systems were treated by Fantz and Wnderlich[7] using the TraDiMo computer program. The latterauthors dealt with a lot of high energy electronic systemsof H2 including some double potential minima electronicsystems, but rotationvibration coupling was not included.

The electronic systems of C2 and O2 are not predomi-nant in the radiative properties of COH2 plasma and werenot investigated in this paper since an accurate descriptionof their radiative properties was proposed by Babou et al.[8] for C2 and by Chauveau [9] for O2. These transitionprobability tables can be widely used to make spectralsimulations for these molecules. We did not find dataconcerning the potential barrier appearing for the thirdpositive system of CO. For this reason, we decided to usethe tables proposed by Babou et al. [8] as they stand.

The triatomic radiative contribution of CO2, H2O andCH4 could predominate at low temperatures, but this topicwill not be developed here since some accurate andextensive databases of line positions and transition prob-abilities exist like CDSD-4000 [10] for CO2, HITEMP [11] forH2O and HITRAN [12] for CH4.

3. Description of the procedure for the calculation of therovibrational transition probabilities

We ignored the spin-splitting and lambda-doubling ofthe rotational levels in the calculation of the rovibrationalwave-functions since, for the molecules involved in thispaper, the rovibrational transition probabilities dependmainly on the coupling between vibrational and purerotational motion. The energy separation of the rotationallevels due to the spin-splitting and lambda-doublingeffects is smaller than the energy separation for twoadjacent pure rotational levels N. Thus the molecular levelswill be simply labelled (assuming Hund's coupling cases bor intermediate a/b) by the electronic n, vibrational v, andpure rotational N quantum numbers. We calculated thetransition probabilities for the P, Q and R rotationalbranches (N0, 71) for all electronic systems, and for

O and S branches (N72) for molecules assumingwed transitions with N72.

Taking into account the rotational dependence of thetransition probabilities, the spontaneous emission Einsteincoefficient Aul for a molecular transition is then usuallyexpressed by the product of the rovibrational transitionprobability pvvNN and the HnlLondon factor as follows:

Aul 64 4s3ul

3hpvvNN SJJ2J1 2

sul is the line position in cm1, pvvNN corresponds to therovibrational transition probability (in a.u.) and SJJ refersto the HnlLondon factors which describe the relativeintensity between each rotational branch for a given total

solving the nuclear radial Schrdinger equation given by

d2v;Nrdr2

22

Ev;N U0rNN12

2r2

!" #v;Nr 0

5

with Ev,N corresponding to the rovibrational energy eigen-values, the reduced mass, U0(r) is the potential energy-curve which is added to the centrifugal term, is thereduced Planck's constant and N is the pure rotationalquantum number.

The preliminary step of this work consisted in thecalculation of the potential-energy curve U0(r) whichcomes into play in Eq. (5). Then the radial Schrdingerequation was solved to calculate the rovibrational wave-functions. Finally, using the most up-to-date electronictransition dipole moment function Re(r) given in theliterature, we calculated the rovibrational transition prob-abilities of interest pvvNN for all the vibrational bands. Thisprocedure was systematically used for all the electronicsystems summarised in Table 2.

3.1. Reconstruction of the potential-energy curves: the RKR

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444436rotational level J. These coefficients have to be normalisedwith the convention proposed by Whiting [13] (as aconsequence it induces the normalisation convention forpvvNN):

JSJJ 20;2S12J1 3

The tables of HnlLondon coefficients for each kind ofelectronic transition (S2S) were summarised byKovcs [14]. The HnlLondon factors closely depend onthe type of Hund's coupling considered (a, b, intermediatea/b) and on the resolved degeneracies of the fine andhyperfine structures of the rotational levels. In this paperwe assumed that all electronic states belong to the b orintermediate a/b Hund's coupling cases. The labelling forthe rotational branches corresponding to the same N aresummarised in Table 1. The rovibrational transition prob-ability pvvNN could then be safely multiplied with thecorresponding HnlLondon factors to perform spectrumcalculations.

The analytical expression for the transition probabilitypvvNN obeys to the relation

pvvNN Z 10

v;Nr Rer v;Nr dr 2

4

Re(r) is the electronic transition moment function (ETMF), v;Nr and v;Nr are the nuclear rovibrational wave-functions for the upper and lower levels respectively and ris the internuclear distance. For more theoretical detailswe invite the reader to refer to the extensive descriptionproposed by Hertzberg [4] and Hougen [15].

A fundamental step of our work was the calculationof the rovibrational wave-functions occurring in (4) by

Table 1Rotational branch labelling for b and a/b Hund's coupling case.

NNN

11 11 22 22, 22 or22

N2 OQ12N1 P P P11,

P22,PQ12P11,P22,PQ12

N0 Q Q11,Q22,QP21,QR12N1 Q R R11,

R22,RQ21R11,R22,RQ21

N2 SR21Molecule Electronicsystem

Upper-lowerstates

ETMF selectedRe(r)

vmax : vmax

CO Infrared X1X1 [16] (49:40)Fourthpositive

A1X1 [17] (22:35)

CO Comet-Tail A2X2 [18] (30:26)Firstnegative

B2X2 [19] (30:35)

BaldetJohnson

B2A2 [19] (30:26)

OH Meinel X2X2 [20] (13:12)Violet A2 X2 [21] (9:13)

CH Infrared X2X2 [22] (12:11)

A2X2 [23] (7:12)

B2X2 [24] (1:12)

C2X2 [24] (2:12)

CH A1X1 [25] (10:17)inversion method

We used the well-known RydbergKleinRees inver-sion procedure for the reconstruction of the rotationlesspotential curves. The original RKR inversion procedure,bearing the name of its main contributors Rydberg [26],Klein [27] and Rees [28] is based upon the WKB (WentzelKramersBrillouin) semi-classical approximation which isimplicitly included in the BornOppenheimer approxima-tion as well as the adiabatic assumption. The procedureconsists in finding the vibrational turning points rinner(v)and router(v) of the internuclear rotationless potential U0(r).

Table 2Electronic molecular systems and electronic dipole moment references.

the calculation of the vibrational wave-functions. Note thatit is not a crucial point since the main information on theoscillatory properties of the wave-functions comes fromthe central part of the potential curve but it couldsignificantly affect the results obtained for the highervibrational levels. As proposed by Chauveau [40], theattractive wing of the potential was extrapolated using aHulburtHirshfelder function, and we used a parametricpower potential for the repulsive wing according to thefollowing expressions:

Urepr arb 10

UHHatt r De1er re23rre3e2r re1rre 11

De is the dissociation energy of the electronic state, re the

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444 437The vibrational G(v) and rotational B(v) energy termswere calculated in the Dunham polynomial expansionformalism as follows:

Gv i 1Yi0 v12

i e v

12

exe v

12

2

6

Bv i 0Yi1 v12

i Bee v12

7

Table 3Molecular spectroscopic constants and vibrational limit of experimentaldata for each electronic state.

Molecule Electronic state Ref. for Dunham Yij Ref. for De vMAXexp

CO X1 [29] [30] 41

A1 [31] [30] 23

CO X2 [32] [19] 5

A2 [32] [19] 8

B2 [33] [19] 2

OH X2 [34] [34] 13

A2 [34] [34] 9

CH X2 [35] [36] 4

A2 [35] [36] 3

B2 [37] [36] 1

C2 [38] [36] 2

CH X1 [39] (curve)

A1 [39] (curve) where Yij are the Dunham polynomial coefficients whichare intimately correlated to the equilibrium spectroscopicconstants. The references for the spectroscopic constantsused for the RKR procedure are summarised in Table 3 foreach electronic state.

The two auxiliary functions introduced by Klein [27] tocalculate the turning points were used as follows:

f v h

82c

s Z vv0

dvGvGv

p 12routerv rinnerv

8

gv 82ch

r Z vv0

Bv dvGvGv

p 12

1rinnerv

1routerv

9where v0 is the value of the vibrational level for which theDunham polynomial became nil.

This method allows a step by step reconstruction of therotationless potential curves U0(r) until the correspondingexperimental vibrational limit of the spectroscopic con-stants used. Above the vibrational limit of the observedexperimental lines, the RKR method is not valid becauseDunham polynomial fit becomes meaningless. It is thennecessary to extrapolate the attractive and repulsive wingsof the RKR potential curve to ensure the convergence ofequilibrium internuclear distance, and a, b, , and arethe adjustable parameters obtained by fitting the lastpoints of each RKR potential wing.

The potential-energy curves were reconstructed withan iteration step of v0.01 until v1, and then withv0.1 until the experimental limit vMAXexp of Table 3. Theintegrals of f(v) and g(v) diverge at the neighbourhood ofthe upper limit, where v approaches v. The convergentpart of the integral was performed by a Simpson procedurewith a resolution of 106 and the divergent part wascalculated using a 96-point Gauss-type quadratureapproximation.

The rotationless potential curves U0(r) obtained by thisprocedure for all the electronic states of interest in thispaper are represented in Figs. 1, 4, 7, 10 and 12. Thesepotentials were now injected in the nuclear radial Schr-dinger Eq. (5) to calculate the energy eigenvalues and therovibrational wave functions required for the calculation ofthe rovibrational transition probabilities.

3.2. Procedure for the calculation of the rovibrational wavefunctions

Using the RKR potential curves U0(r) previouslyobtained, we now have to solve the nuclear radial Schr-dinger Eq. (5). In this work we used the log derivative

Fig. 1. Calculated potential curves for the X1 and A1 electronicstates of CO.

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444438method proposed by Johnson [41] based on the previousworks of Cooley [42] and Blatt [43] to resolve the eigen-value problem. The particular solutions of (5) were

Fig. 2. Comparison of the radiative lifetimes calculated in the presentwork for the A1 electronic states of CO with the references (a) [8], (b)[49], (c) [17], (d) [47] and (e) [48].

Fig. 3. Comparison of the vibrational Einstein coefficients of the X1

electronic states of CO calculated in the present work with the references(a) [50], (b) [16], (c) [8] and (d) [9].

Fig. 4. Calculated potential curves for the X2 , A2 and B2 electronicstates of CO .

Fig. 5. Comparison of the radiative lifetimes calculated in the presentwork for the A2 electronic states of CO with the references (a) [8], (b)[19], (c) [18], (d) [51] and (e) [52].

Fig. 6. Comparison of the radiative lifetimes calculated in the presentwork for the B2 electronic states of CO with references (a) [8], (b)[19], (c) [18], (d) [53] and (e) [54].

Fig. 7. Calculated potential curves for the X2 and A2 electronicstates of OH.

Fig. 8. Comparison of the radiative lifetimes calculated in the presentwork for the X2 electronic states of OH with the references (a) [56], (b)[57] and (c) [58].

Fig. 9. Comparison of the radiative lifetimes calculated in the presentwork for the A2 electronic states of OH with the references (a) [21], (b)[34], (c) [60], (d) [61] and (e) [62].

Fig. 10. Calculated potential curves for the X2; A2; B2 ; and C2

electronic states of CH.

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444 439Fig. 11. Comparison of the radiative lifetimes calculated in the presentwork for the A2 electronic states of CH with the references (a) [64], (b)[65], (c) [23], (d) [66], (e) [67] and (f) [68].calculated using the improved Numerov-type iterationproposed by Simos [44] which upgraded the originalNumerov procedure by extending the interval of periodi-city. Thus, using a small internuclear distance step of1020 m for the integration, the accuracy of this methodcan be considered as suitable.

The vibrational eigenvalues obtained were then com-pared with the experimental energies calculated with theDunham polynomial expansion. Our results are in verygood agreement with the experimental vibrational ener-gies. In the experimental range of vibrational and rota-tional levels the relative deviation of the calculatedrovibrational energy eigenvalues compared with theexperimental ones never exceeds 0.2% (encountered forthe higher vibrational levels), and tends to be 0% for thelower vibrational levels. It validates the accuracy of thecalculated rovibrational wave-functions and thus of thewhole calculation procedure used in this paper. We calcu-lated the complete set of rovibrational wave functionsnv;Nr for all electronic states n involved in the electronicsystems of interest.

Fig. 12. Calculated potential curves for the X1 and A1 electronicstates of CH .

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444440nv;N 1

n;v;NAnvNnvN13

The accuracy of the transition probabilities obtained inthe end closely depends on the accuracy of the wave-functions, but even more on the accuracy of the ETMF usedin the calculation. Large discrepancies exist between thevarious ETMF published in the literature and particularattention has to be paid to select the most reliable. Weexpected the selected ETMF to accurately reproduce thevarious experimental observations and radiative lifetimes.Moreover, we also expect these ETMF to be defined over alarge range of internuclear distances to ensure the descrip-tion of the highest vibrational wave-functions. The ETMFmust absolutely cover the entire internuclear regionwherethe overlapping of the upper and lower wave-functions isnot nil to make an exhaustive description of the transitionprobabilities.

The ab initio calculations of the ETMF are usuallydefined for a large range or internuclear distance but havethe disadvantage of being closely dependent on theelectronic orbital basis set used and large discrepanciesoccur between the various methods used. The experimen-tal ETMF obtained by inversion of the observed lineintensity are quite accurate and, naturally, provide anaccurate reproduction of the experimental radiative life-times of reference but are restricted to the internuclearrange where the molecular lines are observed. So thisdirect experimental inversion method by the r-centroidapproximation cannot apply to the higher vibrationallevels.

For the elaboration of an extensive transition probabil-ity database which includes the high vibrational levels andwhich allows calculations at temperatures up to 15 kK, wesystematically preferred ab initio ETMF rather than theexperimental ones for our calculations. We finally ensuredthat radiative lifetimes obtained for each electronic stateusing ab initio ETMF were in satisfying agreement withexperimental ones. We summarised in Table 2 the selectedETMF used to compute the transition probability tablespublished in Appendix A.

4. Validation of the transition probabilities for theelectronic systems

The set of equilibrium spectroscopic constants used inthe KleinDunham polynomial expansion formalism forthe computation of the energy potentials will not bedetailed more here for each electronic state. We havepragmatically used the more recent set of equilibriumspectroscopic constants based on the largest number of3.3. Calculation of the transition probabilities

Larsson [45] proposed a formalism for the expression ofthe rovibrational Einstein coefficient AvNvN and for theradiative lifetime nv of an electronic state (svv has to beexpressed in cm1 and pvvNN in atomic units):

AnvNnvN 2:026:106sul320;20;

pvvNN 12vibrational levels observed. All the references for thecomputation of the RKR procedure are given in Table 3.

A systematic validation of the rovibrational transitionprobabilities obtained by the procedure described in theprevious sections is difficult since the experimentallyobserved rotational lines depend on many parameters likethe populations of the rotational levels, the line positionsand the line shape. For this reason we have pragmaticallychosen to validate our results by comparing the vibrationallifetimes obtained with our calculations with the experi-mental ones. So ignoring the rotational dependence of thetransition probabilities in a vibrational band, according toLarsson [45], we obtain the vibrational Einstein coeffi-cients Avv and the corresponding vibrational radiativelifetime nv;:

An;vn;v 2:026:106sul320;20;

pvv0;0 14

nv 1

n;vAn;vn;v15

For all the electronic systems summarised in Table 2, wewill briefly justify our choice for ETMF by a critical analysisamong those reported in the literature. Then, the radiativelifetimes and the vibrational Einstein coefficients obtainedusing the selected ETMF were systematically comparedwith the experimental and theoretical results alreadypublished in the literature. Validations for the highestvibrational levels were difficult since the experimentalobservations are restricted to the vibrational levels whererotational lines were observed. We systematically used theab initio ETMF which present the best agreement withexperimental radiative lifetimes and are defined in thelarger range of internuclear distances. Whenever possible,we used unperturbed ETMF which are usable for a largefield of applications since possible corrections could beadded during the spectrum calculation procedure to takeinto account the perturbations using predissociation ratesand perturbed spectroscopic constants. The tables ofvibrational transition probabilities proposed in this paperdo not take into account the perturbations occurring onsome individual vibrational levels and the predissociationand cascade effects possibly affecting the population ofsome vibrational levels. It explains the discrepanciesobserved between our results and some experimentalones for vibrational levels of perturbed electronic states.

4.1. CO electronic systems

The fourth positive system A1X1 of CO greatlycontributes to the discrete radiation of the COH2 plasmain the VUV region and has been thoroughly investigated inthe literature. We retained this electronic system for thevalidation of the whole procedure used in this paper. Usingvarious ETMF, we compared the vibrational Einstein coef-ficients and the calculated radiative lifetimes obtained byour method with those published in the literature. Thebest agreement with the experimental radiative lifetimeswas obtained with the ab initio ETMF of Kirby and Cooper[17] as shown in Fig. 2. Kirby and Cooper explained theincreasing behaviour of their theoretical radiative lifetimes

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444 441versus the decreasing behaviour of the experimental onesby the strong perturbations and predissociation effectscaused by the nearby states. Moreover, some pressure andcascade phenomena strongly affect the exited electronicstates of CO and could make experimental measurementsdifficult. These authors proposed a critical review of theprevious ETMF published to justify the accuracy of theirresults. Finally, the ETMF of Kirby and Copper was con-firmed by the later experimental works of Chan et al. [46].Compared to the transition probabilities published byBabou et al. [8], maximal discrepancies never exceeded3% for the individual Avv (maximal deviation encounteredfor the highest vibrational bands), and less than 0.3% forthe radiative lifetimes. This very weak difference validatesour integral procedure in comparison with the matrixmethod for the resolution of the Schrdinger equationsince we have used the same ETMF. As shown in Fig. 2 weobtained a very good agreement compared with theradiative lifetimes of Kirby and Cooper [26]. Some dis-crepancies appear versus the experimental lifetimes ofField et al. [47] and Imhof and Read [48] but our resultsare still in reasonable agreement with these experimentalresults. The differences are largely explained by the per-turbations affecting the A1 state for the experimentalmeasurements. Our results differ significantly from thoseof Lino Da Silva and Dudeck [49]. As pointed out by Babouet al. [8], this discrepancy is probably explained by thelarge difference between the FranckCondon factors of Ref.[49] and ours.

For the infrared X1X1 system of CO, weretained the ab initio ETMF of Langhoff and Bauschlicher[16] rather than that of Okada et al. [50] since it gives abetter reproduction of the transition probabilities of thewell-established HITRAN database [12] for the fundamen-tal. Nevertheless both give a good reproduction of theexperimental observations for the first and second over-tones. A critical review of the earlier empirical and ab initioETMFs was proposed in Ref. [16]. We did not find experi-mental radiative lifetimes for the ground state of CO butsome Einstein coefficients for the fundamental, first andsecond overtones were available to validate our results. Asshown in Fig. 3, a good agreement was observed comparedto the Einstein coefficients published in Refs. [16,50] andotherwise with those of Babou et al. [8].

4.2. CO electronic systems

We selected the ETMF of Marian et al. [18] for theComet-Tail A2X2 electronic system since it repro-duces the experimental radiative lifetimes a bit better thanthe ETMF of Okada and Iwata [19]. Nevertheless these twoETMF's both gave very similar results for the higher levels.As shown in Fig. 5, a remarkable agreement is observedamong all the theoretical and experimental radiative life-times for the A2 electronic state.

We preferred the ab initio ETMF of Okada and Iwata[19] rather than those of Marian et al. [18] for the firstnegative B2X2 and BaldetJohnson B2A2electronic systems of CO . The more recent ETMF's ofOkada and Iwata have a similar dependence on the inter-nuclear distance in comparison to those of Marian et al.but reproduce the experimental radiative lifetimes a bitbetter for the B2 state. Okada and Iwata. also publishedthe corresponding Einstein coefficients for the two indivi-dual transition systems involving B2 . Our results are ingood agreement with theirs for the first negative systemthough our Einstein coefficients are around twice as highas theirs for the BaldetJohnson system. Since our resultsare close to those of Babou et al. [8] concerning theFranckCondon factors, the discrepancies observed com-pared with Okada and Iwata [19] are possibly caused bythe use of a different degeneracy factor in relation (14). Asshown in Fig. 6, our results are in excellent agreementwith the various experimental lifetimes for the B2 stateof CO .

4.3. OH electronic systems

For the computation of the transition probabilities ofthe Meinel system X2X2, we used the recent ETMF ofVan der Loo and Groenenboom [55] rather than those ofLanghoff and Bauschlicher [20] as it provides a goodreproduction of the experimental radiative lifetimeobserved for the first vibrational level as shown in Fig. 8.

For the 1st positive system A2X2 of OH, the abinitio ETMF of Bauschlicher and Langhoff [21] was used. Itwas validated by the experimental observations of Steffenset al. [59] and then by Luque and Crosley [34]. As shown inFig. 9, the radiative lifetimes obtained using this ETMF arein excellent agreement with the experimental results.

4.4. CH electronic systems

For the infrared electronic system X2X2 calcula-tions, we used the theoretical ETMF of Lie and Hinze [22]rather than the more recent one of Follmeg et al. [63]because the latter was defined on too restrictive inter-nuclear range. We did not find any experimental results tovalidate our transition probabilities. Nevertheless, ourresults are in perfect agreement with those calculated byLie and Hinze [64] using the theoretical transition matrixelements.

We selected the more recent ETMF of Larsson andSiegbahn [23] rather than that of Van Dishoeck [24] forthe first positive electronic system A2X2. Using thisETMF, we obtained an excellent agreement between ourradiative lifetimes for the A2 state compared with theexperimental ones as shown in Fig. 11.

Since the B2 state of CH presents a very smooth curveand a potential barrier due to the high interaction causedby the crossing interaction with the nearby states, we usedthe more recent works of Kalemos et al. [36] obtained byab initio calculations but the height of this potential barrier(811 cm1, located at the distance of 1.85 1010 m)remains a subject of disagreement.

Then we constrained the fitting procedure to make surethat the extrapolated attractive wing of the potentialpasses through the barrier maximum and then smoothlydecreases until the dissociation limit. We used the ab initiotransition moment of Van Dishoeck [69] for second posi-tive system B2X2. This ETMF was normalised tothe experimental lifetime value of 325725 ns [67].

The potential-energy curve of the B2 state is stronglyperturbed by the nearby states and is not well-known,thus it predissociates close to the level v1 causinghigh uncertainties. These perturbations explain the pooragreement between the various results found in theliterature. Under these considerations, our calculationsfor the Einstein coefficient ratios for the B2 state are inexcellent agreement with the experimental results ofLuque and Crosley [37] as shown in Table 4.

The C2 state presents a potential barrier too, due tothe crossing with the nearby state. As for the B2 , thesame procedure was used to describe this potential-energycurve using the potential barrier height of 1630 cm-1

located at 1.7461010 m obtained by Kalemos et al.[36]. The ab initio ETMF of Van Dishoek [69] was usedfor the third positive system C2 X2. The validationof this systemwas made by comparing our results with theintensity ratio of Jeffries et al. [71] as shown in Table 5. TheC2 is strongly predissociative as pointed by [67] and thepurely radiative lifetime has not been observed yet sincethe natural lifetime for this state is too small. Neverthelessour results accurately reproduce the emission intensity

the polynomial form of Dunham. Even though the ground

The aim of this paper was to propose extensive tables ofrovibrational transition probabilities for the main electro-nic systems contributing to the radiation process in COH2plasma. The main electronic systems of CO, CO , OH, CHand CH were investigated in this work. The calculationswere performed using the RKR procedure for the recon-struction of the internuclear potential curve of eachelectronic state of interest. The most accurate spectro-scopic constants were rigorously selected in the Dunhampolynomial formalism to obtain the expression of theenergy levels. Then the radial Schrdinger equation wassystematically solved with an improved Numerov-typemethod to calculate the rovibrational wave-functions.After a critical review of the ETMF published for eachelectronic system, we finally calculated some extensivetables of transition probabilities for the main moleculartransitions occurring in a COH2 plasma. The rotationaldependence of the rovibrational transition probabilities

T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444442state X1 could have been treated by the RKR procedure,we chose to use the curves proposed by Helm et al. [39] asthe energy potential for the two states. These ab initio

Table 4Comparisons of our calculated Einstein emission coefficient ratios to theexperimental results of Luque and Crosley [37] and Jeffries et al. Garland

and Crosley [70] for the B2X2 transition of CH.

Ratio This work Luque et al. Garland et al.

A01=A00 0.028 0.03170.005 0.0670.02A10=A11 0.87 0.8570.08 0.570.2A12=A11 0.164 0.16570.02 0.370.1

Table 5Comparisons of our calculated Einstein emission coefficient ratios to the

experimental results of Jeffries et al. [71] for the C2X2 transition ofCH.

Ratio This work Jeffries et al.

A01=A00 0.0066 0.005370.001A02=A01 0.020 0.01570.006A10=A11 0.0123 0.009570.005A12=A11 0.0060 0.005170.0014A10=A12 2.04 2.071.1A13=A12 0.018 o0.05ratio determined experimentally by Jeffries et al. [71]. Anexcellent agreement was also found with the ab initioradiative lifetimes obtained by Hinze et al. [64].

4.5. CH electronic system

The electronic systems of CH have not been exten-sively studied in the past. As noted by Carrington andRamsay [72], the A1 state is strongly perturbed by cross-ing interactions. The RKR procedure became useless forthis state since the vibrational energies cannot be fitted topotentials seem very accurate and were confirmed by themeasurements. We used the well accepted ETMF of Lars-son and Siegbahn [25] for our calculations since it accu-rately reproduces the recent experimental observation.Comparisons for the radiative lifetimes obtained werequite difficult since we only found experimental valuesfor the first level. We decided to validate our results bycomparison of our absorption oscillator strength withthose published in the more recent works of the literatureas shown in Table 6. As expected, our results are inexcellent agreement with the theoretical results of Larssonand Siegbahn, which tends to validate the potential curvesused. We also observed an excellent agreement with theexperimental oscillator strength of Larsson and Siegbahn[25] but some small discrepancies appear compared withWeselak et al. [73] when v increases. We think thatprobably either the dipole moment of Larsson and Sieg-bahn is not correct at large internuclear distances or themeasurements of Weselak et al. are not accurate for thehigher vibrational levels. Anyway the observed discrepan-cies are not critical since the oscillator strengths for themost intense bands are in remarkable agreement.

5. Conclusion

Table 6

Calculated band oscillator strengths (x105) for the A1X1 transitionof CH .

f(v,v) This work Larsson et al. [25] ExperimentalWeselak et al. [73]

(0,0) 545 545(0,1) 105 108(1,0) 325 331 34276(1,1) 89 88(1,2) 101 100(1,3) 28 29(2,0) 138 1729(3,0) 54 7578(4,0) 21 4075(2,1) 242 246(3,1) 216 211

php [20.04.13.].[7] Fantz U, Wnderlich D. FranckCondon factors, transition probabil-

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Appendix A

The tables of rovibrational transition probabilities aregiven under electronic form and can be found in the onlineversion of this article. The individual columns of thesupplementary data tables refer respectively to the upperv and lower v vibrational levels, the upper N and lowerN pure rotational levels, and the transition probabilitypvvNN.

Appendix A. Supporting information

Supplementary data associated with this article can befound in the online version at http://dx.doi.org/10.1016/j.jqsrt.2013.09.005.

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T. Billoux et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 133 (2014) 434444444

Tables of radiative transition probabilities for the main diatomic molecular systems of OH, CH, CHplus, CO and COplus...IntroductionBackgroundDescription of the procedure for the calculation of the rovibrational transition probabilitiesReconstruction of the potential-energy curves: the RKR inversion methodProcedure for the calculation of the rovibrational wave functionsCalculation of the transition probabilities

Validation of the transition probabilities for the electronic systemsCO electronic systemsCOplus electronic systemsOH electronic systemsCH electronic systemsCHplus electronic system

ConclusionAppendix ASupporting informationReferences