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The spectroscopic parameters of sodium cyanide, NaCN (X˜1A’), revisited HolgerS.P.Mu¨llera,∗,D.T.Halfenb,L.M.Ziurysb aI.PhysikalischesInstitut,Universita¨tzuKo¨ln,Zu¨lpicherStr.77,50937Ko¨ln,Germany bDepartmentsofChemistryandAstronomy,ArizonaRadioObservatoryandStewardObservatory,UniversityofArizona,933NorthCherryAvenue,Tucson, AZ85721,USA Abstract 2 The studyofthe rotationalspectrumofNaCN (X˜1A’)hasrecentlybeenextendedin frequencyandin quantumnumbers. Dif- 1 0ficultieshavebeenencounteredin fittingthetransitionfrequencieswithinexperimentaluncertainties. Varioustrialfitstracedthe 2difficultiestotheincompletediagonalizationoftheHamiltonian. Employingfewerspectroscopicparametersthanbefore,thetran- sition frequenciescould be reproducedwithin experimentaluncertainties on average. Predictionsof a-type R-branchtransitions n awith Ka ≤ 7 up to 570 GHz should be reliable to better than 1 MHz. In addition, modified spectroscopicparametershave been Jderivedforthe13CisotopicspeciesofNaCN. 3 Keywords: ]rotationalspectroscopy,circumstellarmolecule,centrifugaldistortion,Hamiltoniandiagonalization A G .1. Introduction promptingHalfen and Ziurys [5] to study the rotational spec- h trumofNaCNbetweenapproximately180and530GHz. They p - There are severalways to form a moleculefrom a metal M encounterteddifficultiesinreproducingtheirexperimentaltran- oand the CN group. Severalmetals, such as thetransitionmet- sitionfrequenciesonaveragewithinexperimentaluncertainties r tals copper and zinc, prefer a linear cyanide arrangementwith andattributedthistothefloppynatureofthemolecule. s an MC bond. Linear isocyanides with an MN bond are pre- In the course of creating an updated catalog entry for the a [ferredbyotherssuchasthemaingroupmetalsmagnesiumand CologneDatabaseforMolecularSpectroscopy(CDMS)[6,7], aluminum.Sodiumandpotassium,however,displayadifferent thepreviouslyreportedexperimentaldataforNaCN[1,5]were 1 structuralmotif,anearlyT-shapedstructurewithalmostequally criticallyevaluated.Usingfewerspectroscopicparametersthan v 3longMCandMNbonds.Asaconsequence,bothmoleculesare before, it was possible to reproduce both data sets well if the 0asymmetrictoprotors. Hamiltonianwasdiagonalizedsufficiently. 7 The rotational spectrum of NaCN has been studied by van 0 Vaals et al. [1] between 9 and 40 GHz employing molecular- 1.beamelectric-resonancespectroscopy. 20rotationaltransitions 2. Considerationsforthespectralanalysis 0 were recorded, half of them being much weaker b-type tran- 2 One findsratherfrequentlythattheset of spectroscopicpa- sitions. NaCN was detected in the circumstellar envelope of 1 rameters, which reproduces a given list of transition frequen- :the carbon-richstar CW Leo, also knownas IRC +10216[2], v cieswithinexperimentaluncertainties,isnotunique. Oneway soonafterthreepairsofrelatedunidentifiedlines,alsodetected i of reducing the ambiguity of a parameter set is to reduce the Xin that source, had been attributed to MgNC [3]. Recently, rmserror,alsoknowsastheweightedrms,ofthefitwithasfew rHe et al. [4] described a molecular line survey of CW Leo at a parametersas possible and reasonable. A reasonable parame- 2 and 1.3 mm. In order to analyze the NaCN emission, the ter setrequirescertainlowerordertermto be usedbeforeone spectroscopic parameters of NaCN were reevaluated because ofhigherorderisallowedtobeincluded;e.g.,theinclusionof initially, thequarticcentrifugaldistortionparameterswere ex- L in the fit requiresthat at least one of the two parameters pressed as τ values, which are rarely implemented in current JJK H or H has already been used in the fit. In the absence of spectroscopy programs. The predictions derived from the re- J JK strong correlation among the parameters, it suffices to search sulting parameters were good enough for the NaCN emission for the one which reduces the rms error the most. However, features obtained in the course of that line survey as well as suchcorrelationisrathercommon,suchthatsearchesforsmall for previously published transitions. Observations at slightly parametersetsmaybemorecomplex. Itisfrequentlyassumed higherfrequenciesandwithabettersignal-to-noiseratio,how- thatthe smallest parameterset, which reproducesthe parame- ever, revealed increasing deviations largely with increasing J, terset best, willprovidethemostreasonablepredictions. Itis useful in this context to address the issue of uncertainties at- ∗Correspondingauthor. tributedtothetransitionfrequencies. Besidesappropriateesti- Emailaddress:[email protected](HolgerS.P.Mu¨ller) mates and, less frequently in recent years, no estimates at all, PreprintsubmittedtoJournalofMolecularSpectroscopy January4,2012 onealsofindsuncertaintieswhichseemtobetoooptimisticor Veryrecently, HalfenandZiuryspublishedan extensiveset toopessimistic, leadingtoweightsinenlargeddatasetswhich of NaCN transition frequencies all pertaining to the strong a- aretoohighortoolow,respectively,iftheseestimatesstayun- typeR-branchtransitionswith 11 ≤ J′′ ≤ 32and0 ≤ K ≤ 6 a corrected.Ifthelistoftransitionfrequenciesissufficientlylarge between 179 and 531 GHz [5]. In order to provide an up- with respecttothe numberofspectroscopicparametersvaried dated entry for the CDMS [6, 7], these data were subjected inthefit,anrmserrorof1.0overallaswellasforwelldefined to combined fits together with the previous data [1]. Since sub-setsofthefrequencylistcanbeviewedasanindicationof HalfenandZiurysencountereddifficultiesinreproducingtran- reasonable uncertainty estimates, though caution is advised if sitionsmainlywithhigherK values,itwasdecidedtoaddthe a a very small set of the lines depends particularly strongly on newtransitionfrequencieswitheachK seriesseparatelyinthe a one or few parameters. An additionalcriterion for reasonable present analysis. Moreover, a tight rejection threshold of 3σ uncertaintyestimatesis thedistributionof theresiduals. Most wasappliedsinceitseemedplausiblethatlargeresidualsofone of the lines, about two third, should fit within the uncertain- or few lines may have caused the difficulties. Pickett’s SPCAT ties,buttheremaybesomelineswithresidualsexceedingthree andSPFIT programs[11]wereusedforpredictionandfittingof timestheuncertainties,however,theirnumbershoulddecrease thespectra, respectively,ashasbeendonein thetwoprevious very rapidly with the magnitude of the ratio between residual investigations[4,5]. and uncertainty. It is possible that the overall uncertainty es- StartingwiththeparametersetofRef.[4],theprevioustran- timates in a line list are too pessimistic, but one or few lines sition frequenciesplus12 new oneswith K = 0 could be re- a haveresidualswhicharetoolarge. Arejectionthresholdisap- producedoverallwithinuncertaintiesbyaddingh andomitting 2 plied in finalfits in particularin largerdata sets as a smallset H andh .Atthisstage,onlyoneofthepreviouslinesdeviated K 3 of transition frequencies with large residuals may complicate fromthecalculatedfrequencybyslightlymorethanthreetimes the determination of higher order parameters. A threshold of theuncertainty.Inlaterfits,theamountandidentityofthelines 3σisrathercommon,seee.g. Ref.[8,9],butotherthresholds, displayinglargerresidualchangedsometimes. usuallylessstrict,mayalsobefound. ThesequentialadditionoftransitionfrequencieshavingK = a 1−5 affordedone new parametereach to be included. These were h , h , L , l , and L . Theadditionof K = 6lines 1 3 JJK 4 KKJ a 3. Spectralanalysis didnotrequireanynewparameter. Trialfitswith H included K inthefityieldedavalueof41±55Hzwithchangesinthere- Asmentionedintheintroduction,NaCNaswellasKCNare mainingparameterswithintheuncertainties. Theuncertainties asymmetric tops, in contrast to most other molecules formed of the predominantly K-dependent terms increased markedly, fromametalatomandtheCNgroup.SinceforboththeNatom butinnocasebymorethanafactorof2. SinceHK wasnotde- isclosertothemetalatom[1,10],theyshouldbeviewedasiso- terminedwithsignificance,itwasomittedfromthefinalfit. Itis cyanidesratherthancyanides. However,followingtheconven- noteworthythattheuncertaintyisafactorof4smallerthanthe tion,wewillcontinuecallingthesemoleculescyanides. Ray’s uncertaintyderivedinRef.[4]fromthemicrowavedataonly. asymmetryparameterκ is −0.9567forNaCN, the moleculeis This parameter set is smaller than the one in Ref. [5], but thusfairlyclosetotheprolatesymmetriclimit. Ithasalargea- reproduces almost all of the data within the rejection thresh- dipolemomentcomponentofµ = 8.85Dandamuchsmaller old. Two of the new lines had residuals around200kHz, and a b-componentof∼0.2D1[4].Inspiteofthesmallb-component, one line deviated from its calculated position by more than van Vaals et al. reported10 b-typetransitions with ∆J = ±1, 300kHz. Mostof theotherlinesseemedtohavebeenjudged ∆K =∓1,and∆K =±1alongwith10a-typetransitionswith conservatively with an uncertainty of 50 kHz because transi- a c ∆J = 0,±1, ∆Ka = 0, and ∆Kc = ±1; the quantum numbers tion frequencies with Ka ≤ 3 gave partial rms errors of less extendedtoJandKaof16and3,respectively.Theb-typetran- than 0.5; the values for Ka = 4 and 6 were around 0.6, and sitionscontainedessentiallyallofthe∆Ka , 0informationin onlylineswithKa = 5yieldedapartialrmserrorof1.19. The thatdatasetaswellasbyfarthemost∆J , 0informationbe- previousdata[1]nowyieldedapartialrmserrorof1.17,indi- causemostofthea-typetransitionsfollowed∆J = 0selection cating that the previously reported uncertainty estimates have rules. Inspectionoftheir residualssuggestedthatthe reported been corrected slightly too far. But since none of these lines uncertaintiesareratherconservative;theywereassumedto be showed particularly large residuals with respect to the uncer- 3σ uncertainties in Ref. [4]. Employing thirteen parameters tainties,noneoftheselineswereomittedandtheuncertainties and the adjusted uncertainties, the latter authors were able to wereleftunchanged. reproduce these lines with an rms error of 0.840. The fairly ThefinalsetofspectroscopicparametersisgiveninTable1 large number of parameters compared to the number of lines togetherwithNaCNandKCNvaluesfromRef.[4]andresults makeitlessclearwhethertheadjusteduncertaintiesofthetran- fromafitoftheNaCNmicrowavedatatotheKCNparameter sition frequencieswere estimated appropriately. Nevertheless, set. Thelatterfithas2fewerparametersthantheonepublished, thedatapermittedgoodtoreasonablepredictionswellintothe but has a larger rms error of 1.055 comparedwith 0.840; this millimeterwaveregion. was the reason to discard this fit and to choose the one which requiredmoreparameters. The list of transition frequencies with assignments, uncer- 1http://www.astro.uni-koeln.de/cgi-bin/cdmsinfo?file=e049510.cat tainties, and residuals between experimental frequencies and 2 Table1:Spectroscopicparametersa(MHz)ofNaCN(X˜1A’)fromthepresentcombinedfitincomparisontofitsemployingmicrowavedataonlyandincomparison toKCNalongwithdimensionlessrmserrorsofeachfit. Parameter NaCN,combinedfit Na13CN NaCN,microwavedataonly KCN Present Present Ref.[4] Present Ref.[4] A−(B+C)/2 50101.74547(110) 48021.93283(314) 50101.74757(166) 50101.74606(131) 53528.08107(268) (B+C)/2 7820.189927(145) 7652.415049(391) 7820.190252(259) 7820.190351(206) 4737.751389(93) (B−C)/4 274.1524216(124) 272.7985489(396) 274.1525055(177) 274.1525120(172) 100.959727(64) DK×103 194.11(25) 268.89(142) 196.52(169) 193.56(69) 956.79(247) DJK×103 800.06(2) 738.90(39) 800.23(34) 800.24(19) 374.609(50) DJ×103 13.3123(5) 13.2279(180) 13.3361(160) 13.3313(96) 5.24462(209) d1×103 −2.34537(23) −2.42239(112) −2.34908(34) −2.34914(33) −0.51452(78) d2×103 −1.407340(56) −1.379543(295) −1.408951(182) −1.405461(226) −0.237559(199) HK×103 − − 1.428(220) − − HKJ×106 327.4(18) 303.5(55) 155.9(184) 314.5(28) 101.2(11) HJK×106 −26.31(8) −23.43(30) −3.54(120) −26.72(19) −7.28(11) HJ×109 −3.77(34) −3.5b 0.54(5) − − h1×109 −35.68(17) −34.0b − − − h2×109 −25.40(9) −24.6b − −47.65(199) −5.46(74) h3×109 19.34(9) 19.1b 39.92(688) − − LKKJ×109 −0.384(40) −0.331b − − − LJJK×109 −0.845(43) −0.759b − − − l4×1012 −0.587(80) −0.575b − − − rmserror 0.746 0.807 0.840 1.055 0.827 aNumbersinparenthesesareonestandarddeviationinunitsoftheleastsignificantfigures. bKeptfixedtovaluesfromthecombinedNaCNfitmultipliedwithratiosoftheappropriatepowersofA−(B+C)/2,B+C,andB−C,seesection3. thosecalculatedfromthefinalsetofspectroscopicparameters both H and H were released in the fit in addition to the KJ JK is given in the Supplementarymaterial. Predictionsof the ro- lower order parameters. The resulting fit is also given in Ta- tationalspectrumalongwithadocumentationwillbeprovided ble1.UpdatedpredictionsforNa13CNaswellasauxiliaryfiles in the catalog section2 of the CDMS. Line, parameter, and fit willbeprovidedintheCDMS. files, alongwithfurtherauxiliaryfiles, willbe availableinthe archivesection3. 4. Discussion Ref.[4]alsoprovidedspectroscopicparametersforNa13CN. Sincepredictionsforthemainisotopicspeciesfromthatstudy Itmaybesurprisingthatinthepresentstudyabetterfitwas turnedouttobeunsatifactoryintheuppermillimeterandlower obtained with 17 parameters which is less than 23 employed submillimeterwave regions,it is rather likely thatthisapplies previously [5]. The reason turned out to be rather mundane. to the predictions for Na13CN also. 17 rotational transitions TheprogramsSPFITandSPCATrequireminimumandmaximum hadbeenstudiedbyvanVaalsetal.[1]. Thequoteduncertain- valuesforK tobespecified. Inaprolatetype,asymmetrictop ties for the derived hyperfine free transition frequencies were molecule,theminimumvalueisobviously0,andthemaximum assumed to be 3σ values, as has been done for the main iso- value has to be at least as large as the highest K value in the a topolog. Centrifugal distortion parameters were estimated by line list. The numerical diagonalization of the Hamiltonian, multiplyingtheNaCNvaluesfromthecombinedfitwithratios however,usuallyrequiresthemaximumvaluetobesomewhat oftheappropriatepowersofA−(B+C)/2,B+C,andB−C; largerthanthatvalue. Itappearsasifanincreasebythehigh- e.g.,HKJ withtheratioof(A−(B+C)/2)2(B+C)andh2with estKaorderoftheparametersinthefitisareasonableestimate theratioof(B+C)(B−C)2. Whilethisscalingcannotbeex- for this increase. Transitions in the NaCN line list extend to pected to hold strictly, it is usually a reasonable asssumption, K = 6, and the parameter l connects levels differing in K a 4 a see e.g. Ref. [4] for isotopicspecies of SiC2 and forNa13CN by 8, e.g. the Ka = 4 asymmetry components. Hence, the or Ref. [12] for the example of the isotopic species of vinyl maximumK valuerequestedinthefitshouldbearound14. In cyanide. fact, no change in parameters and in the rms error occurs for Anrmserrorof1.518wasachievedbyjustvaryingtherota- K ≥ 15. The data set for SO in its groundvibrationalstate a 2 tionalandquarticcentrifugaldistortionparameters.Significant extendstoK =28[8];trialfitshaveshownthatamaximumK a reduction of the rms error below values of 1.0 were obtained valueofatleast33shouldberequested.Evenlargermaximum wheneitherHKJ,HJK,HJ,orh3 werevariedinaddition,how- K values may have to be requested for predicting a spectrum ever, the changes for the latter two appeared to be too large atroomtemperatureforconvergenceofthecalculatedpartition in comparisonto the respectivevaluesand were discarded. A function. slightlybetterfitwithanrmserrorof0.807wasobtainedwhen AscanbeseeninTable1,thelowerorderspectroscopicpa- rameters, up to the quartic centrifugal distortion terms, agree 2http://www.astro.uni-koeln.de/cdms/catalog very well with the previous parameter determined from mi- 3http://www.astro.uni-koeln.de/site/vorhersagen/catalog/archive/ crowavedataonly. Sincemerelyselectedparametersofhigher 3 order were employed, in particular for the MW fit, it is prob- transition frequenciesdiffer by only 60 MHz. Transitions ap- ably not surprising, that the agreement for the higher order proaching oblate pairing contribute to the determination of parameters is not so good. In the alternative MW data fit, purely K-dependentparameters. At still higher J, these tran- which employs exactly the parameters which were used in sitionswillreachoblatepairingandtransitionswithK = J−1 c the KCN fit, the parameters H and H agree well with etc. willapproachandeventuallyreachoblatepairingalso. In KJ JK the present values, and only h differs by almost a factor of fact,theuncertaintyof H inpresenttrialsfitswasreducedto 2 K two. It is interesting to note, that this fit yields much bet- 55 Hz (see section 3), a factor of 4 smaller than the value of ter predictions of the new data [5] than the one published 220 Hz obtained from microwave data only [4], see also Ta- earlier [4]. The 19 − 18 transition, e.g., predicted by ble1, indicatingthatthenewlymeasreda-typeR-branchtran- 0,19 0,18 the alternative MW fit at 284160.864 (461) MHz was mea- sitions[5]haveimprovedthepurelyK-dependentparameters. sured at 284161.701 (50) MHz, almost within twice the pre- Obviously, ∆K , 0 transitions probe the K-level structure a dicted uncertainty. The previously published fit [4] predicts moredirectly. Inthatcase,itmaybemorepromisingtosearch 284169.538(634) MHz. Apparently, the parameter set which for ∆K = 2 transitions which are frequently much stronger a yieldsthesmallestrmserrorisnotalwaysthebestone,andthis thanb-typetransitions,inparticularinthemillimeterwavere- situation is more likely to occur if the number of parameters gion. Transitions with K ≤ 6 and not too high J should be a iscomparativelylargewithrespecttothenumberoftransition predictedsufficientlywellwiththe largestdeviationsfromthe frequenciesorifacertainparameterisdeterminedbyonlyfew predictedfrequenciesoforderof10MHz. transitions. Thecentrifugaldistortionparameterswhichhavebeenvaried The present parameter set not only reproduces the experi- intheNa13CNfitareclosetotheinitialvaluesobtainedbyscal- mental transitions well, but it should also permit to a certain ing NaCN values. The most pronounceddeviation occurs for degreereliablepredictionsbeyondtheexperimentallyaccessed D , whichshows a verylargeisotopic shift. Such large shift, K quantum numbers. It should be pointed out that the discus- however, can be attributed to the accidentially small value of sionofthepredictivepowerofthepresentparametersetiscon- D comparedwithD andisnotuncommon.Alargeisotopic K JK cernedmainlywitha-typetransitionsbecausethea-component shift was also observed, e.g., for ∆ of 37Cl35ClO compared K 2 of8.85Disverymuchlargerthantheb-componentof∼0.2D. to 35Cl35ClO which was fully accountedfor by the harmonic 2 Theb-typetransitionsareweakerthanthea-typetransitionsby forcefieldcalculation[13].Arelativelymuchsmallerdeviation (µ /µ )2 ≈ 1960, making them unimportant for radioastron- occursford ,whosemagnitudeforthe13Cisotopoicspeciesis a b 1 omy. Nob-typetransitionshavebeenprovidedintheprevious actuallyslightlylargerthanthatofthemainspecies. Itispos- CDMScatalogentry,andthiswillalsoapplyforthenewentry. siblethatthisunexpectedrelationiscausedbytheassumptions We consider a prediction to be reliable if the frequency of madeforsomeofthefixedhigherorderparameters. a transition not yet measured will be found to deviate from Predictions of the rotational transition frequencies of thepredictedfrequencybyanamountgenerallynotexceeding Na13CNshouldbeviewedwithmuchgreatercautionthanthose three timesthepredicteduncertaintyandneverexceedingthat forthemainisotopicspeciesbecauseofthemuchsmallerdata value by far. Assuming that the effects of centrifugal distor- set. Nevertheless,thepredictionsshouldbereliabletoreason- tion parameters not included in the fit do not exceed the pre- ableatleastupto J of15,maybe20,anduptoK of4,maybe a dicteduncertainties,predictionsmaybereliableaslongasthe 5. uncertainties do not exceed 1 MHz. At low values of K this Halfen and Ziurys [5] derived a rotational temperature of a correspondsto J upto around50. Extrapolationin K isvery about41KforNaCNinthecircumstellarenvelopeofCWLeo. a J-dependent. Around J = 15, K = 10 maybepredictedrea- The Boltzmannpeak atthis temperatureoccursslightlyabove a sonably, around J = 35, K = 9 is the limit, and beyond J of 200GHz,andthelineintensitiesappeartobenegligibleabove a 40, even predictionswith K = 8 should be viewed with cau- 500 GHz. Moreover, transitions with K = 10 are around 3 a a tion. Sincetheassumptionmentionedaboveisnotnecessarily ordersofmagnitudeweakerthanthosewith K = 0or1. The a justified, all extrapolations should be viewed with additional predictionsof the NaCN rotationalspectrumderivedfrom the caution. presentstudyshouldthusbeadequateforallobservationalpur- TheparameterH maybethemostimportantonenotyetin- poses,atleastasfarasthegroundvibrationalstateisconcerned. K cludedinthefitwhichwillhaveeffectsonthecalculatedtran- Thebendingmode(3 =1)istheexcitedvibrationalstatelow- 2 sitionfrequenciesandenergies. UsingavalueofH =200Hz est in energy. It is ∼170 cm−1 or ∼245 K above ground [1]. K asfixed,avaluealmostfourtimestheuncertaintyof H from Transitionspertainingtothisstatewillbehardtodetectinthe K trial fits, see section 3, and thus near the upper end of pos- absenceofinfraredpumping,which,however,cannotberuled sible values, changes the energies by less than 1 cm−1 for out. J = K ≤ 21. The a-type transitions frequencies change by a insignificantamountsfortheJ andK rangeforwhichthepre- a 5. Conclusions dictionsaredeemedtobereliable,seeabove. The easiest way to improve the K-level structure of NaCN TherotationalspectrumofNaCNhasbeenreanalyzed. The will be recording higher-J a-type R-branch transitions with previousdifficultiesinreproducingtheexperimentaltransition ∆K =0becauseoftheirfavorableintensities.TheK = Jtran- frequencies [5] have been attributed to the incomplete diago- a c sitionarealreadyapproachingoblatepairingastheJ =33−32 nalizationoftheHamiltonian. Thepredictionsgeneratedfrom 4 thepresentdatasetshouldbegoodenoughforallobservational purposesrestricted to the groundvibrationalstate. The modi- fiedparametersetforNa13CNshouldprovidebetterprediction oftherotationalspectrumuptomoderatequantumnumbers. Acknowledgements H.S.P.M. is very gratefulto the Bundesministeriumfu¨rBil- dung und Forschung (BMBF) for financial support aimed at maintainingtheCologneDatabaseforMolecularSpectroscopy, CDMS. This support has been administered by the Deutsches Zentrumfu¨rLuft-undRaumfahrt(DLR).L.M.Z.acknowledges supportfromNSFgrantAST-09-06534. AppendixA.Supplementarymaterial Supplementary data for this article are available on Sci- enceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://library.osu.edu/sites/msa/jmsahp.htm). Supplementary dataassociatedwiththisarticlecanbefound,intheonlinever- sion,atdoi:. References [1] J.J. van Vaals, W.L. Meerts, A. Dymanus, Chem. Phys. 86 (1984) 147−159. 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